whose the front and back squares are coCartesian. Applying the pasting law of pushouts iteratively, we find that $\varphi$ turns out to be cobase change of the map \eqref{eq:latchingmap1}, which is a cofibration. We therefore get the conclusion.
\end{proof}

The category of simplicial $C$-collections admits a canonical simplicial model structure. It implies that the category $()$ admits a canonical simplicial model structure, by the transferred principle in simplicial version (see [[43], Propositions 3.1.5, 3.2.8]). One constructs the Hochschild resolution of $$ as follows.

\begin{cons}  Let $\sH_\bullet\P : \Delta^{\op} \lrar \BMod(\mathcal{P})$ be the simplicial object of $\P$-bimodules with $\sH_n\P := \P^{\circ(n+2)}$, the face map $d_i : \sH_n\P\lrar \sH_{n-1}\P$ given by using the composition $\mu : \P\circ\P \lrar \P$ to combine the factors $i+1$ and $i+2$ in $\sH_n\P$ and with the degeneracy map $s_i$ given by inserting the unit operations of $\P$ between the factors $i+1$ and $i+2$. The realization $\diag(\sH_\bullet\P) \in \BMod(\mathcal{P})$ has $n$-simplices being those of  $\sH_n\P$. The  map $\mu$ induces a canonical map of simplicial objects $\sH_\bullet\P \lrar \P$. The \textbf{augmentation map} $\psi : \diag(\sH_\bullet\P) \lrar \diag(\P)=\P$ is then a weak equivalence by [[43], Corollary 3.7.6], (this even comes with a contracting homotopy). The map $\psi$ now exhibits $\diag(\sH_\bullet\P)$ as the \textbf{Hochschild resolution of} $\P\in \BMod(\P)$.

\noindent On other hand, since $\P\circ\P = \sH_0\P$, there is a unique map of simplicial objects $\P\circ\P \lrar \sH_\bullet\P$, which is the identity on degree 0. Now, the diagonal functor gives a map $ \rho : \P\circ\P \lrar \diag(\sH_\bullet\P)$ of $\P$-bimodules, satisfying that the composition $ \P\circ\P \x{\rho}{\lrar} \diag(\sH_\bullet\P) \x{\psi}{\lrarsimequ} \P$ agrees with $\mu : \P\circ\P \lrar \P$.
\end{cons}

\begin{lem}\label{c:acofibration} Suppose that $\P$ is a $\Sigma$-cofibrant simplicial operad. The map $\psi$ indeed exhibits $\diag(\sH_\bullet\P)$ as a cofibrant resolution for $\P$ regarded as a bimodule over itself. Moreover, the map $\rho : \P\circ\P \lrar \diag(\sH_\bullet\P)$ is a cofibration of $\P$-bimodules. In particular, $\diag(\sH_\bullet\P)$ is also a cofibration resolution for $\P$ when regarded as a pointed $\P$-bimodule.
\end{lem}
\begin{proof} The first statement is an analogue of [[43], Corollary 3.7.6]. We now recall his arguments. It suffices to show that the latching map $\rL_{n-1}\sH_\bullet\P \lrar \sH_n\P$ is a cofibration for every $n\geqslant 0$. The case in which $n=0$ is clear since $\sH_0\P = \P\circ\P$ is cofibrant as a $\P$-bimodule. When $n\geqslant 1$, the object $\rL_{n-1}\sH_\bullet\P$ can be written as $\rL_{n-1}\sH_\bullet\P = \P \circ K_{n-1} \circ \P$ where $K_{n-1}$ is the object of [[43], Lemma 3.7.8]. Moreover, there is a canonical map $k_n : K_{n-1} \lrar \P^{\circ n}$ such that the map $\rL_{n-1}\sH_\bullet\P \lrar \sH_n\P$ agrees with the free $\P$-bimodule map generated by $k_n$. It therefore suffices to show that $k_n$ is a cofibration of symmetric sequences for every $n\geqslant 1$. This can be done using an inductive argument. Indeed, note that the map $k_1$ is the same as the unit map $\I_C \lrar \P$, which is a cofibration by the assumption that $\P$ is $\Sigma$-cofibrant. Furthermore, by [[43], Lemma 3.7.8] the map $k_{n+1}$ turns out to be the pushout-product of $k_n$ with the unit map $\I_C\lrar \P$. Finally, one will need the fact that, given two maps $f$ and $g$ of symmetric sequences in a sufficiently nice symmetric monoidal model category, the pushout-product of $f$ with $g$ is a cofibration as soon as both of them are one and, in addition, the domain of $g$ is cofibrant (cf. [[12],  Lemma 11.5.1]).

To prove that the map $\rho$ is a cofibration, we make use of Lemma \ref{l:diagcofib}. Since $\P\circ\P$ is considered as a constant simplicial object, the latching map \eqref{eq:latchingmap} is simply $\rL_{n-1}\sH_\bullet\P \lrar \sH_n\P$ when $n\geqslant 1$ and the identity map $\Id_{\P\circ\P}$ when $n=0$. But the first map is a cofibration by the above paragraph. The proof is hence completed.
\end{proof}

\begin{rem}\label{r:hochschildresalg} Let $\P$ be any simplicial operad and let $A$ be any $\P$-algebra. \textbf{Hochschild resolution of} $A$ is the realization of the simplicial $\P$-algebra $\sH^{\P}_\bullet A$  with $\sH^{\P}_n A = \P^{\circ(n+1)}\circ A$. The augmentation map $\diag(\sH^{\P}_\bullet A) \lrar A$ is a weak equivalence by [[43], Corollary 3.7.4], and indeed exhibits $\diag(\sH^{\P}_\bullet A)$ as a cofibrant resolution for $A$. To see this, we will show that the latching map $\rL_{n-1}\sH^{\P}_\bullet\A \lrar \sH_n^{\P}\A$ is a cofibration for every $n\geqslant 0$. The case $n=0$ is clear since $\sH^{\P}_0\A = \P \circ A$ the free $\P$-algebra generated by $A$. Let us now pick up the objects $K_{n}$'s and the maps $k_n$'s mentioned in the above proof.  When $n\geqslant1$ the map $\rL_{n-1}\sH^{\P}_\bullet\A \lrar \sH^{\P}_n\A$ concides with the free $\P$-algebra map generated by $k_n\circ A : K_{n-1}\circ A\lrar \P^{\circ n} \circ A$. It therefore suffices to show that the map $k_n\circ A$ is injective for every $n\geqslant 1$.  The key point is that, in the monoidal category of simplicial symmetric sequences, pushout-product of any two injections is again an injection (cf. [[43], Proposition 3.4.5]). By an inductive argument we can show that the maps $k_n$ are injective, and hence the maps $k_n \circ A$ are all injective as well.
\end{rem}

\begin{proof}[\underline{Proof of Proposition~\ref{p:splcsetS9}}]  In the first step, we follow the arguments of [[22], section 5]. By applying the functor $\E$ to $\sH_\bullet\P$ degreewise, one obtains a simplicial object, $\E \sH_\bullet\P$, of operads under $\mathcal{P}\sqcup\mathcal{P}$. The realization $\diag(\E \sH_\bullet\P)$ is then an operad under $\mathcal{P}\sqcup\mathcal{P}$. Since $\E(\P) \cong \P$ in $\Op_C(\Set_\Delta)_{\P\sqcup\P/}$, there is a canonical map $\varphi_\P : \diag(\E \sH_\bullet\P) \lrar \P$ of operads under $\mathcal{P}\sqcup\mathcal{P}$.
One observes that there is a canonical isomorphism $\diag(\E \sH_\bullet\P)\cong \E(\diag(\sH_\bullet\P))$ of operads under $\mathcal{P}\sqcup\mathcal{P}$ and over $\P$ (cf. [[22], Proposition 5.3]). Since $\E(\diag(\sH_\bullet\P))$ is already a model for $\LL \E(\P)$ by Lemma \ref{c:acofibration}, it just remains to show that the map $ \varphi_\P : \diag(\E \sH_\bullet\P) \lrar \P$ is a weak equivalence of operads. By the diagonal principle, $\varphi_\P$ is a weak equivalence as soon as the map $\E \sH_n\P\lrar \P$ is one for every $n\geqslant0$. Moreover, one  finds that $\E \sH_n\P \cong \P \sqcup \F_*(\P^{\circ n}) \sqcup \P$ where $$\F_* : \Coll_C(\Set_\Delta)_{\I_C/} \lrar \Op_C(\Set_\Delta)$$ refers to the free-operad functor on \textit{pointed $C$-collections}. This tells us that if $\Q\lrarsimeq\P$ is a weak equivalence between cofibrant operads then $\varphi_\Q$ is a weak equivalence if and only if $\varphi_\P$ is one. Applying the diagonal principle in the other direction, we get that $\varphi_\P$ is a weak equivalence as soon as the map $\varphi_{\P_{(n)}}$ is one for every $n\geqslant0$ (where $\P_{(n)}$ is the  operad of $n$-simplices of $\P$).

Now, consider $\P$ as an $\textbf{O}_C$-algebra with $\textbf{O}_C$ being the operad of simplicial $C$-colored operads. The above remark suggests that we can make use of $\P^{c} := \diag(\sH^{\textbf{O}_C}_\bullet \P)$ as (another) cofibrant model for $\P$. By the first paragraph, it suffices to verify that the map $\varphi_{\P^{c}}$ is a weak equivalence. To this end, we just need to show that the map $\varphi_{\P^{c}_{(n)}}$ is a weak equivalence for every $n$. More precisely, we have that $$\P^{c}_{(n)} = ((\textbf{O}_C)^{\circ(n+1)} \circ \P)_{(n)} = (\textbf{O}_C)^{\circ(n+1)} \circ \P_{(n)} .$$ (The second identification is because of the fact that $\textbf{O}_C$ is a discrete operad). In particular, $\P^{c}_{(n)}$ is a discrete free $\textbf{O}_C$-algebra. Note that a free $\textbf{O}_C$-algebra is the same as the free operad generated by a \textit{free symmetric sequence} (i.e., symmetrization of a nonsymmetric sequence).

By the second paragraph, we can assume without  loss of generality that $\P$ is the free operad generated by a discrete free symmetric sequence or alternatively, $\P$ is the symmetrization of a discrete free nonsymmetric operad. Nevertheless, for the remainder we just need to assume that $\P$ is the symmetrization of a nonsymmetric operad $\Q$, i.e., $\P =\Sym(\Q)$ (see \S \ref{s:operad}). Back to the first paragraph, we therefore have to show that the map $ \varphi_{\Sym(\Q)} : \diag(\E \sH_\bullet\Sym(\Q)) \lrar \Sym(\Q)$ is a weak equivalence of operads. This is in fact equivalent to our original problem: proving that $\LL\E\Sym(\Q) \cong \Sym(\Q)$.

Observe that the symmetrization functor lifts to the functors $\Sym : \BMod(\Q)^{*} \lrar \BMod(\Sym(\Q))^{*}$ and $\Sym : \nsOp_C(\Set_\Delta)_{\mathcal{Q}\sqcup\mathcal{Q}/} \lrar \Op_C(\Set_\Delta)_{\Sym(\Q)\sqcup\Sym(\Q)/}$, which are left adjoints to the associated forgetful functors. Moreover, we have a commutative square of left Quillen functors
$$ \xymatrix{
\BMod(\Q)^{*} \ar^{\E}[r]\ar_{\Sym}[d] & \nsOp_C(\Set_\Delta)_{\mathcal{Q}\sqcup\mathcal{Q}/} \ar^{\Sym}[d] \\
\BMod(\Sym(\Q))^{*} \ar_{\E}[r] & \Op_C(\Set_\Delta)_{\Sym(\Q)\sqcup\Sym(\Q)/} \\
(for this, it suffices to verify the commutativity of the associated square of right adjoints). According to [[22], Proposition 5.4], we have that $\LL\E\Q \cong \Q$. Thus, by the commutativity of the above square and by the fact that the symmetrization functor preserves weak equivalences, we obtain the expected identification $\LL\E\Sym(\Q) \cong \Sym(\Q)$.
\end{proof}

\begin{rem}\label{r:hochschildresModR} Hochschild resolutions work in the context of simplicial $R$-modules under a slightly different setting. Let $\P\in \Op_C(\sMod_R)$ be given such that the unit map $\I_C \lrar \P$ is a cofibration of symmetric sequences. Then the composition $\P\circ\P\lrar\diag(\sH_\bullet\P)\lrarsimeq \P$ exhibits $\diag(\sH_\bullet\P)$ as a cofibrant resolution for $\P\in \BMod(\mathcal{P})^{*}$. On other hand, let $A$ be a levelwise cofibrant $\P$-algebra. Again, the augmentation map $\diag(\sH^{\P}_\bullet A) \lrarsimeq A$ exhibits $\diag(\sH^{\P}_\bullet A)$ as a cofibrant resolution for $A$. For the proof, one repeats the arguments given in the proof of Lemma \ref{c:acofibration} and Remark \ref{r:hochschildresalg}.
\end{rem}

\begin{prop} \label{p:splcmoduleS9} The category $\sMod_R$ satisfies the condition (S8) \ref{S8}. Namely, for every cofibrant object $\P \in \Op_C(\sMod_R)$, the left derived functor $\LL\E:\BMod(\mathcal{P})^{*}\lrar \Op_C(\sMod_R)_{\mathcal{P}\sqcup\mathcal{P}/}$ of $\E$ sends $\P$ to itself $\mathcal{P}$.
\end{prop}
\begin{proof} By the above remark, we can make use of $\diag(\sH_\bullet\P)$ as a cofibrant resolution for $\P\in \BMod(\mathcal{P})^{*}$ and $\diag(\sH^{\textbf{O}_C}_\bullet \P)$ as (another) cofibrant model for $\P\in \Op_C(\sMod_R)$, (note that since the operad $\textbf{O}_C$ is discrete $\Sigma$-cofibrant, its unit map is indeed a cofibration). We then pick up the first two paragraphs of the proof of Proposition \ref{p:splcsetS9}. We hence may assume that $\P$ is the free operad generated by a discrete free symmetric sequence.

The free-forgetful adjunction $R\{-\} : \adjunction*{}{\Set_\Delta}{\sMod_R}{} : \mathbb{U}  $ lifts to a Quillen adjunction $$R\{-\} : \adjunction*{}{\Op_C(\Set_\Delta)}{\Op_C(\sMod_R)}{} : \mathbb{U} $$ between operads. By the assumption on $\P$, there exists a simplicial operad $\Q\in \Op_C(\Set_\Delta)$ which is the free operad generated by a discrete free symmetric sequence such that $R\{\Q\} = \P$. On other hand, the functor $R\{-\}$ does lift to a left Quillen functor $R^{b}\{-\} : \BMod(\mathcal{Q})^{*} \lrar \BMod(\mathcal{P})^{*}$, which fits into the following commutative square of left Quillen functors
$$ \xymatrix{
\BMod(\mathcal{Q})^{*} \ar^{\E}[r]\ar_{R^{b}\{-\}}[d] & \Op_C(\Set_\Delta)_{\mathcal{Q}\sqcup\mathcal{Q}/} \ar^{R\{-\}}[d] \\
\BMod(\mathcal{P})^{*} \ar_{\E}[r] & \Op_C(\sMod_R)_{\mathcal{P}\sqcup\mathcal{P}/} \\
Note that both the functors $R^{b}\{-\}$ and $R\{-\}$ preserve weak equivalences. Now, by the commutativity of the square we have that $$\LL R \{\LL\E(\Q)\} \cong \LL\E (\LL R^{b}\{\Q\}) = \LL \E (\P) .$$
On other hand, Proposition \ref{p:splcsetS9} proves that $\LL\E(\Q) \cong \Q$, and hence we get $\LL \E (\P) \cong \LL R \{\Q\} \cong \P$ as expected.
\end{proof}

\smallskip

\subsection{Cotangent complex of enriched operads}\label{s:cotangentcplx}

\smallskip

Let $$ be an $$-enriched $C$-colored operad.

\begin{notns} To avoid confusion, in the remainder of this section, we will use the symbol $\bigsqcup$ (resp. $\underset{b}{\bigsqcup}$, $\underset{pb}{\bigsqcup}$, $\underset{c}{\bigsqcup}$) standing for the coproduct operation in the category $\Op(\mathcal{S})$ (resp. $\BMod(\mathcal{P})$, $\BMod(\mathcal{P})^{*}$, $\Op_C(\cS)$).
\end{notns}

\begin{notns}\label{no:cotangentcomplex} We let $\rL_\P\in \T_\P \Op(\mathcal{S})$ and $\rL_\P^{b}\in \T_\P \BMod(\mathcal{P})$, respectively, denote the \textbf{cotangent complexes of} $\P$ when regarded as an object of $\Op(\mathcal{S})$ and $\BMod(\mathcal{P})$. Besides that, we denote by $\rL^{\red}_\P\in \T_\P \Op_C(\mathcal{S})$ the cotangent complex of $\P$ when regarded as an object of $\Op_C(\mathcal{S})$ and refer to it as the \textbf{reduced cotangent complex of} $\P$.
\end{notns}

\begin{conv}\label{conv:reducedQcohom} From now on, by \textbf{Quillen cohomology of} $\P$ we will mean the Quillen cohomology of $\P$ when regarded as an object of $\Op(\mathcal{S})$, which is therefore classified by $\rL_\P$. On other hand, by \textbf{reduced Quillen cohomology of} $\P$ we will mean the Quillen cohomology of $\P$ when regarded as an object of $\Op_C(\mathcal{S})$, which is classified by $\rL^{\red}_\P$.
\end{conv}

By Theorem \ref{t:mainoptangent}, when $𝒫$ is fibrant and $Σ$-cofibrant, we have a sequence of Quillen equivalences connecting the tangent categories:
$$ \adjunction*{\simeq}{\T_\P\IbMod(\P)}{\T_\P\BMod(\P)}{} \adjunction*{\simeq}{}{\T_\P\Op_C(\mathcal{S})}{} \adjunction*{\simeq}{}{\T_\P\Op(\mathcal{S})}{} .$$

\begin{notns}\label{no:severaladj} We will write $\adjunction*{\F^{ib}_{\P}}{\T_\P\IbMod(\P)}{\T_\P\Op(\mathcal{S})}{\U^{ib}_{\P}}$ and $\adjunction*{\F^{b}_{\P}}{\T_\P\BMod(\P)}{\T_\P\Op(\mathcal{S})}{\U^{b}_{\P}}$ standing for two composed adjunctions taken from the above sequence.
\end{notns}

In order to get the desired formula of Quillen cohomology of $$, we will compute the derived image of $_∈_(𝒮)$ under the composed right Quillen equivalence $$ \U^{ib}_{\P} : \T_\P \Op(\mathcal{S}) \overset{\simeq}{\longrightarrow} \T_\P\BMod(\P) \overset{\simeq}{\longrightarrow} \T_\P \IbMod(\mathcal{P}) .$$
As the first step, we will show that the derived image of $_$ in $_()$ is weakly equivalent to $_^b$, up to a shift. Our work therefore extends [[9], Proposition 3.2.1], but in a different approach. For our approach, the base category $$ is technically required to satisfy the extra condition (S8) \ref{S8}. After having done that first step, it just remains to compute the derived image of $_^b$ in $_(𝒫)$. Let us see how it goes.

As discussed above, we first wish to prove the following.

\begin{prop}\label{cotantgentcplxOperadandBimodule} Suppose that $\mathcal{S}$ additionally satisfies the condition (S8) \ref{S8} and that $\P$ is fibrant and $\Sigma$-cofibrant. Then the  left Quillen equivalence $\F_\P^{b} : \T_\P \BMod(\mathcal{P}) \lrarsimeq \T_\P \Op(\mathcal{S})$ identifies $\rL^{b}_{\mathcal{P}}$ to $\rL_{\mathcal{P}}[1]$. Alternatively,  the right  Quillen equivalence $\U_\P^{b} : \T_\P \Op(\mathcal{S}) \lrarsimeq \T_\P \BMod(\mathcal{P})$ identifies $\rL_{\mathcal{P}}$ to $\rL^{b}_{\mathcal{P}}[-1]$.
\end{prop}

The proof of Proposition \ref{cotantgentcplxOperadandBimodule} will require several technical lemmas.

Given two $$-enriched categories $$ and $$, the tensor product $⊗$ is by definition the $$-enriched category whose set of objects is $(⊗) := () ×()$ and such that for every $c,c'∈()$ and $d,d'∈()$ we have
$$ \Map_{\C\otimes\D}((c,d),(c',d')) := \Map_\C(c,c') \otimes \Map_\D(d,d') .$$
Recall that the category of $$-bimodules is isomorphic to $(^⊗,𝒮)$ the category of $$-valued enriched  functors on $^⊗$. Under this identification, the functor $_: ^⊗𝒮, (x,y) ↦_(x,y)$ is exactly $$ viewed as a bimodule over itself.

\begin{lem}\label{cotantcplxofenrichedcat} Let $\mathcal{C}\in \Op(\mathcal{S})$ be a fibrant and levelwise cofibrant operad concentrated in arity 1. Then there is a weak equivalence $\theta_\C : \F^{b}_\C(\rL^{b}_{\mathcal{C}}) \lrarsimeq  \rL_{\mathcal{C}}[1]$ in $\T_\C \Op(\mathcal{S})$.
\end{lem}
\begin{proof} We also regard $\C$ as an $\cS$-enriched category. The proof is then straightforward by observing that the category $\Cat(\mathcal{S})$ is already a $``$neighborhood'' of $\mathcal{C}$ in $\Op(\mathcal{S})$.  This idea is expressed as follows.  There is  a commutative square of left Quillen functors
$$ \xymatrix{
\T_{\Map_\C}\Fun(\C^{\op}\otimes\C,\mathcal{S}) \ar[r]\ar_{\F^{b}_{\Map_\C}}[d] & \T_\C \BMod(\mathcal{C}) \ar^{\F^{b}_{\C}}[d] \\
\T_\C \Cat(\mathcal{S}) \ar[r] & \T_\C \Op(\mathcal{S}) \\
The horizontal functors are the obvious embedding functors, which clearly preserve cotangent complexes, while the functor $\F^{b}_{\Map_\C}$ is the left Quillen equivalence appearing in [[9], Theorem 3.1.14]. According to  Proposition 3.2.1 of loc.cit, there is a weak equivalence $\ovl{\theta}_\C : \F^{b}_{\Map_\C}(\rL_{\Map_{\mathcal{C}}}) \lrarsimeq \rL_{\mathcal{C}}[1]$ in $\T_\C \Cat(\mathcal{S})$. Finally, the expected weak equivalence $\theta_\C$ is given by the image of $\ovl{\theta}_\C$ under the embedding functor $\T_\C \Cat(\mathcal{S}) \lrar \T_\C \Op(\mathcal{S})$.
\end{proof}

In what follows, we consider the case where $=_C$ the initial $C$-colored operad and describe the weak equivalence $θ__C : ^b__C(^b__C) __C[1]$ of the above lemma.

Let us pick up several notations of [[9], \S 3.2]. We denote by $*$ the category which has a single object whose endomorphism object is $1_$.  Moreover, let $[1]_$ denote the category with objects $0, 1$ and mapping spaces $_[1]_(0,1) = 1_, _[1]_(1,0) = ∅$ and $_[1]_(0,0) = _[1]_(1,1) = 1_$. Localizing $[1]_$ at the unique non-trivial morphism $0 1$ gives us the category $[1]_^∼$, which is the same as $[1]_$ except that $_[1]^∼_(1,0) = 1_$. By construction, the canonical map $[1]^∼_*$ is a weak equivalence.

Take a factorization $[1]_ℰ [1]_^∼$ of the canonical map $[1]_[1]_^∼$ into a cofibration followed by a trivial fibration. We now obtain a sequence of maps
$** [1]_ℰ [1]_^∼ *$
such that the first two maps are cofibrations, while the others are weak equivalences. Tensoring with $_C$ (viewed as an $$-enriched category) produces a sequence of maps in $(𝒮)$
$$ \I_C \bigsqcup \I_C \lrar \I_C \otimes [1]_\cS \lrar \I_C \otimes \mathcal{E} \lrarsimeq  \I_C \otimes [1]_\cS^{\sim} \lrarsimeq \I_C.$$
The last two maps are again weak equivalences, while the others are again cofibrations because $_C$ is discrete.

By the above words, the pushout $_C   _C ⊔_C   _C⊗ℰ$ is a cofibrant model for $_C   _C ⊔_C   _C ∈()__C//_C$, and hence $__C[1]$ is given by
$$ \rL_{\I_C}[1] \x{\defi}{=} \Sigma^{\infty}(\I_C \, \underset{\I_C \sqcup \I_C}{\x{\h}{\bigsqcup}} \, \I_C) \simeq \Sigma^{\infty}(\I_C \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C\otimes \mathcal{E}) .$$

On the other hand, by definition $^b__C$ is given by $Σ^∞(_C   b   _C)$. Furthermore, note that the category $(_C)$ is isomorphic to $_C()$ the category of $C$-collections. So we find that $^b__C(^b__C) = Σ^∞((_C))$ where $$ is the free functor $_C() _C()$ and $(_C)$ is regarded as an object in $(𝒮)__C//_C$. In fact, $(_C)$ is the same as the pushout $_C _C⊔_C_C ⊗[1]_$. Finally, we find the expected map
\begin{equation}\label{eq:thetaIC}
\theta_{\I_C} : \F^{b}_{\I_C}(\rL^{b}_{\I_C})  = \Sigma^{\infty}\left(\I_C \underset{\I_C\sqcup\I_C}{\bigsqcup}\I_C \otimes [1]_\cS \right) \lrarsimeq \Sigma^{\infty}\left(\I_C \underset{\I_C\sqcup\I_C}{\bigsqcup}\I_C \otimes \mathcal{E} \right) = \rL_{\I_C}[1]
\end{equation}
canonically induced by the map $[1]_ℰ$.

\smallskip

Consider $μ: ∘$ as a map in $(𝒫)$ and the unit map $η: _C $ of $$ as a map in $(𝒮)$. Recall by notation that $Σ^∞_+(μ)$ is the image of $μ$ under the left Quillen functor $Σ^∞_+ : ()_/ _(𝒫)$. Note that $Σ^∞_+(μ)$ has already the right type since $∘∈(𝒫)$ is cofibrant. Also, $Σ^∞_+(η)$ is the image of $η$ under the left Quillen functor $Σ^∞_+ : (𝒮)_/ _(𝒮)$, and has already the right type as well.

\begin{lem}\label{l:muandeta} There is a weak equivalence $\F_\P^{b}(\Sigma^{\infty}_+(\mu)) \lrarsimeq \Sigma^{\infty}_+(\eta)[1]$ in $\tpop$.
\end{lem}
\begin{proof} The map $\eta : \I_C \lrar \P$ gives rise to a commutative square of left Quillen functors
\begin{equation}\label{eq:lbplp}
\xymatrix{
\T_{\mathcal{I}_C}\BMod(\mathcal{I}_C)   \ar^{\F_{\I_C}^{b}}[r]\ar_{\eta^{b}_!}[d] & \T_{\mathcal{I}_C}\Op(\mathcal{S}) \ar^{\eta^{op}_!}[d] \\
\T_\P \BMod(\mathcal{P})\ar_{\F_\P^{b}}[r] & \T_\P \Op(\mathcal{S}) \\
\end{equation}
Let us start with the cotangent complex $\rL^{b}_{\mathcal{I}_C}\in \T_{\mathcal{I}_C}\BMod(\mathcal{I}_C)$ of $\I_C \in \BMod(\mathcal{I}_C)$. Note that the functor $\BMod(\mathcal{I}_C) \lrar \BMod(\mathcal{P})$ agrees with the free $\P$-bimodule functor $\Coll_C(\cS)\lrar \BMod(\mathcal{P})$, which in particular takes $\I_C$ to $\P\circ\P$. Due to this, we find that the functor $\eta^{b}_!$ sends $\rL^{b}_{\mathcal{I}_C}=\Sigma^{\infty}(\mathcal{I}_C \, \underset{b}{\bigsqcup} \, \mathcal{I}_C)$ to $\Sigma^{\infty}(\P \, \underset{b}{\bigsqcup} \, (\P\circ\P))\in \T_\P \BMod(\mathcal{P})$, which is exactly $\Sigma^{\infty}_+(\mu)$. Thus, the commutativity of \eqref{eq:lbplp} shows that $\F_\P^{b}(\Sigma^{\infty}_+(\mu)) = \eta^{op}_! \F_{\I_C}^{b}(\rL^{b}_{\mathcal{I}_C})$. On other hand, by the words after Definition \ref{d:relcotangent} we have $\Sigma^{\infty}_+(\eta)[1] = \eta^{op}_!(\rL_{\mathcal{I}_C}[1])$. Using $\Sigma^{\infty}(\I_C \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C\otimes \mathcal{E})$ as a cofibrant model for $\rL_{\mathcal{I}_C}[1]$ as discussed above, we find the desired weak equivalence given by
$$ \F_\P^{b}(\Sigma^{\infty}_+(\mu)) = \eta^{op}_! \F_{\I_C}^{b}(\rL^{b}_{\mathcal{I}_C}) \x{\eta^{op}_!(\theta_{\I_C})}{\underset{\simeq}{\xrightarrow{\hspace*{1.3cm}}}} \eta^{op}_!(\rL_{\mathcal{I}_C}[1]) = \Sigma^{\infty}_+(\eta)[1] $$
where $\theta_{\I_C}$ is the weak equivalence \eqref{eq:thetaIC}.
\end{proof}

\begin{rem}\label{r:necess} It is necessary to give an explicit description of the map $\eta^{op}_!(\theta_{\I_C})$. Concretely, we have that $$\F_\P^{b}(\Sigma^{\infty}_+(\mu)) = \eta^{op}_! \F_{\I_C}^{b}(\rL^{b}_{\mathcal{I}_C}) = \Sigma^{\infty} \left ( \P \, \underset{\I_C}{\bigsqcup} \, \Fr(\I_C) \right ) = \Sigma^{\infty} \left ( \P \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes [1]_\cS \right ).$$
On the other hand, $$ \Sigma^{\infty}_+(\eta)[1] = \eta^{op}_!(\rL_{\mathcal{I}_C}[1]) = \Sigma^{\infty} \left ( \P \, \underset{\I_C}{\bigsqcup} \; (\I_C \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C\otimes \mathcal{E}) \right ) = \Sigma^{\infty} \left ( \P \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes \mathcal{E} \right ) .$$
We now find the map $\eta^{op}_!(\theta_{\I_C})$ given by applying $\Sigma^{\infty}$ to the map
$$ \P \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes [1]_\cS \lrar \P \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes \mathcal{E}$$
canonically induced by the map $[1]_\cS \lrar  \mathcal{E}$.
\end{rem}

Consider the Quillen adjunction $ℒ^_: *__C(𝒮)_(𝒮) : ℛ^_$.

\begin{lem}\label{CotangentCplxOpC&OP} The left Quillen functor $\mathcal{L}^{\Sp}_\P$ takes $\rL^{\red}_{\P} \in \T_\P \Op_C(\mathcal{S})$  to $\rL_{\P/\I_C}$ the relative cotangent complex of the unit map $\eta : \I_C \lrar \P$ (cf. $\S$\ref{s:tangentcategory}).
\end{lem}
\begin{proof} Let $\P^c \lrarsimeq \P$ be a cofibrant resolution of $\P$ in $\Op_C(\mathcal{S})$. So we get a factorization $\I_C  \lrar \P^c \lrarsimeq \P$ in $\Op(\mathcal{S})$ of the map $\eta$ into a cofibration followed by a weak equivalence. By Remark \ref{r:relcotant} we have that $\rL_{\P/\I_C} = \Sigma^{\infty}(\P \, \underset{\I_C}{\bigsqcup} \, \P^c)$. On other hand, by definition the reduced cotangent complex $\rL^{\red}_{\P} \in \T_\P \Op_C(\mathcal{S})$ is given by $\Sigma^{\infty}(\P \, \underset{c}{\bigsqcup} \, \P^c)$ the suspension spectrum of $\P \, \underset{c}{\bigsqcup} \, \P^c \in \Op_C(\mathcal{S})$ considered as a $C$-colored operad over and under $\P$.

Observe now that since $\I_C$ is initial object in $\Op_C(\mathcal{S})$, the coproduct $\P \, \underset{c}{\bigsqcup} \, \P^c$ is nothing but the pushout $\P \, \underset{\I_C}{\bigsqcup} \,\P^c$  when regarded as an object in $\Op(\cS)$. We hence get the conclusion.
\end{proof}

Suppose that $$ is cofibrant. Then $_/_C$ is simply given by $Σ^∞(  _C   )$. Nevertheless, we will need two more models for this. Let $$ and $'$ be in $()$ with $$\R = \P \, \underset{\I_C}{\bigsqcup} \, (\I_C \otimes [1]_\cS) \; \; , \; \;  \R' = \P \, \underset{\I_C}{\bigsqcup} \, (\I_C \otimes \mathcal{E}) .$$
We then form a diagram of coCartesian squares in $()$ as follows
\begin{equation}\label{eq:pushoutsss}
\xymatrix{
\I_C \ar[r]\ar_{i_0}[d] & \P \ar[d] \\
\I_C \, \bigsqcup \, \I_C   \ar[r]\ar[d] & \P \, \bigsqcup \,  \I_C  \ar[r]\ar[d] & \P \, \bigsqcup \, \P \ar[d] \\
\I_C \otimes [1]_\cS   \ar[r]\ar[d] & \R  \ar[r]\ar[d] & \R \, \underset{\I_C}{\bigsqcup} \, \P \ar[d] \\
\I_C \otimes \mathcal{E}  \ar[r]\ar_{\simeq}[d] & \R'  \ar[r]\ar_{\simeq}[d] & \R' \, \underset{\I_C}{\bigsqcup} \, \P \ar_{\simeq}[d] \\
\I_C \ar[r] & \P \ar[r] & \P \, \underset{\I_C}{\bigsqcup} \, \P \\
\end{equation}
The middle column is nothing but the image of the first column through the left adjoint functor $()__C//_C ()_//$ induced by the unit map $η: _C $. We consider $$ and $'$ as objects in $()_//$ via that way. Three squares on the right hand side are considered as coCartesian squares in $()_//$. Moreover, note that all the arrows in this diagram are cofibrations, except the three bottom vertical maps, which are all weak equivalences (the last two ones are homotopy cobase change of the weak equivalence $_C ⊗ℰ _C$).

\begin{lem}\label{l:twomodels} The map $\R \lrar \R'$ induces a weak equivalence $\theta_{\P/\I_C} : \Sigma^{\infty}(\R \, \underset{\I_C}{\bigsqcup} \, \P) \lrarsimeq \Sigma^{\infty}(\R' \, \underset{\I_C}{\bigsqcup} \, \P)$ of spectrum objects in $\T_\P \Op(\mathcal{S})$. Moreover, the two are both weakly equivalent to the relative cotangent complex $\rL_{\P/\I_C}$.
\end{lem}
\begin{proof} By the weak equivalence $\R' \, \underset{\I_C}{\bigsqcup} \, \P \lrarsimeq \P \, \underset{\I_C}{\bigsqcup} \, \P$ of objects in $\Op(\cS)_{\P//\P}$ as indicated above, the suspension spectrum $\Sigma^{\infty}(\R' \, \underset{\I_C}{\bigsqcup} \, \P)$ is indeed a model for $\rL_{\P/\I_C}$. It remains to show that $\theta_{\P/\I_C}$ is a weak equivalence. For this, we will follow the arguments given in the proof of [[9], Proposition 3.2.1]. It hence suffices to show that the map $\R \, \underset{\I_C}{\bigsqcup} \, \P \lrar \R' \, \underset{\I_C}{\bigsqcup} \, \P$ is (-1)-cotruncated in $\Op(\cS)_{\P//\P}$. Since the latter is homotopy cobase change in $\Op(\cS)_{\P//\P}$ of the map $\R \lrar \R'$, it now suffices to show that $\R \lrar \R'$ is (-1)-cotruncated in $\Op(\cS)_{\P//\P}$. Furthermore, since the map $\R \lrar \R'$ agrees with the image of the map $\I_C \otimes [1]_\cS \lrar \I_C \otimes \mathcal{E}$ through the left Quillen functor $\Op(\cS)_{\I_C//\I_C} \lrar \Op(\cS)_{\P//\P}$, the proof will be hence completed after showing that the latter map is a (-1)-cotruncated map (between cofibrant objects) in $\Op(\cS)_{\I_C//\I_C}$. For this, it suffices to show that the fold map $$\I_C \otimes \mathcal{E} \, \underset{\I_C \otimes [1]_\cS}{\bigsqcup} \, \I_C \otimes \mathcal{E}  \lrar \I_C \otimes \mathcal{E}$$ is a weak equivalence in $\Op(\cS)_{\I_C//\I_C}$ or in $\Cat(\cS)$, alternatively. This follows by the fact that the fold map $\mathcal{E} \, \underset{[1]_\cS}{\bigsqcup} \, \mathcal{E}  \lrar  \mathcal{E}$ is a weak equivalence in $\Cat(\cS)$, due to the convention that $\cS$ satisfies the invertibility hypothesis (cf. $\S$\ref{s:convention}).
\end{proof}

We will denote by $^b_𝒫/𝒫∘𝒫∈_(𝒫)$ the relative cotangent complex of the map $μ: ∘$ regarded as a map in $(𝒫)$. Take a factorization $𝒫∘𝒫 ^ 𝒫$ in $(𝒫)$ of the map $μ$ into a cofibration followed by a weak equivalence. By Remark \ref{r:relcotant} we have that $Σ^∞(∘^)$ is a (cofibrant) model for $^b_𝒫/𝒫∘𝒫$.

\begin{lem}\label{l:relcotanbimod} Suppose that $\mathcal{S}$ additionally satisfies the condition (S8) \ref{S8} and that $\P$ is cofibrant. Then the left Quillen functor $\F_\P^{b} : \T_\P \BMod(\mathcal{P}) \lrar \tpop$ sends $\rL^{b}_{\mathcal{P}/\mathcal{P}\circ \mathcal{P}} \in \T_\P \BMod(\mathcal{P})$ to $ \rL_{\P/\I_C}[1]$.
\end{lem}
\begin{proof} Let us write $\F_\P^{b}$ as the composed left Quillen functor
$$ \T_\P \BMod(\mathcal{P}) \lrar \T_\P \Op_C(\mathcal{S}) \x{\mathcal{L}^{\Sp}_\P}{\lrar} \tpop . $$

As the first step, we will compute the image of $\rL^{b}_{\mathcal{P}/\mathcal{P}\circ \mathcal{P}}$ through the first functor. It will suffice to compute the image of $\P\underset{\P\circ \P}{\bigsqcup}\P^{\cof}$ through the left Quillen functor $\BMod(\P)_{\P//\P} \lrar \Op_C(\mathcal{S})_{\P//\P}$, which is the same as the composition
$$ \BMod(\P)_{\P//\P} \cong \BMod(\P)^{*}_{\P//\P} \x{\widetilde{\E}}{\lrar}  (\Op_C(\mathcal{S})_{\P \, \underset{c}{\bigsqcup} \, \P /})_{\P//\P} \cong  \Op_C(\mathcal{S})_{\P //\P} $$
where $\widetilde{\E}$ is the one lifted by the functor $\E : \BMod(\P)^{*} \lrar \Op_C(\mathcal{S})_{\P \, \underset{c}{\bigsqcup} \, \P /}$ (see $\S$\ref{sub:extracondition}). Note that the pushout $\P\underset{\P\circ \P}{\bigsqcup}\P^{\cof}$, when regarded as an object in $\BMod(\P)^{*}$, is exactly $\P \, \underset{pb}{\bigsqcup} \, \P^{\cof}$ the  coproduct of $\P$ with $\P^{\cof}$ as pointed $\P$-bimodules. Writing $\E(\P^{\cof}) := (\P \, \underset{c}{\bigsqcup} \, \P \lrar \Q)$, we then have that the underlying $C$-colored operad of $\E(\P \, \underset{pb}{\bigsqcup} \, \P^{\cof})$ is given by the pushout $\Q \, \underset{\P \, \underset{c}{\bigsqcup} \, \P}{\bigsqcup} \, \P$. The condition (S8) \ref{S8} implies that $\Q$ is a cofibrant resolution of $\P$ when considered as an object in $\Op_C(\mathcal{S})_{\P \, \underset{c}{\bigsqcup} \, \P /}$ and hence, the latter pushout is a model for $\Sigma(\P \, \underset{c}{\bigsqcup} \, \P)$ the suspension of $\P \, \underset{c}{\bigsqcup} \, \P$ considered as an object in $\Op_C(\mathcal{S})_{\P // \P}$. So we find that the derived image of $\rL^{b}_{\mathcal{P}/\mathcal{P}\circ \mathcal{P}}$ in $\T_\P \Op_C(\mathcal{S})$ is given by $\Sigma^{\infty}(\Sigma(\P \, \underset{c}{\bigsqcup} \, \P))$, which is nothing but $\rL^{\red}_\mathcal{P}[1]$ the suspension of $\rL^{\red}_\P$.

Finally, we deduce by using Lemma \ref{CotangentCplxOpC&OP}, which proves that the derived image of $\rL^{\red}_\mathcal{P}[1]$ through the functor $\T_\P \Op_C(\mathcal{S}) \lrar \tpop$ is exactly $\rL_{\P/\I_C}[1]$.
\end{proof}

\begin{proof}[\underline{Proof of Proposition~\ref{cotantgentcplxOperadandBimodule}}] We can take $f : \Q \lrarsimeq \P$ to be a bifibrant resolution of $\P$ in $\Op(\mathcal{S})$ such that $f$ is a map in $\Op_C(\mathcal{S})$ (cf. Observations \ref{ob:simpleobs}). The map $f$ gives rise to a commutative square of left Quillen equivalences
\begin{equation}\label{eq:WLOGsigmacofib}
\xymatrix{
\T_\Q \BMod(\mathcal{Q}) \ar[r]^{\F_\Q^{b}}_{\simeq}\ar[d]_{\simeq} & \T_\Q \Op(\mathcal{S}) \ar[d]^{\simeq} \\
\T_\P \BMod(\mathcal{P})\ar[r]_{\F_\P^{b}}^{\simeq} & \T_\P \Op(\mathcal{S}) \\
\end{equation}
(cf. Proposition \ref{refineoperad&infinitesimalbimod}). It is then not hard to see that the vertical functors preserve cotangent complexes. Therefore, if the statement holds for $\Q$ then it holds for $\P$ as well. So we can assume without loss of generality that $\P$ is bifibrant.

By the definition of relative cotangent complex, we have two cofiber sequences in $\T_\P \Op(\mathcal{S})$:
$$ \F_\P^{b}(\Sigma^{\infty}_+(\mu)) \lrar \F_\P^{b}(\rL^{b}_{\mathcal{P}}) \lrar \F_\P^{b}(\rL^{b}_{\P / \P \circ \P}) \; \; , \; \; \Sigma^{\infty}_+(\eta) \lrar \rL_{\mathcal{P}} \lrar \rL_{\mathcal{P}/\mathcal{I}_C} .$$
Let us consider the corresponding shifted cofiber sequences:
$$ \F_\P^{b}(\rL^{b}_{\P / \P \circ \P})[-1] \x{\alpha}{\lrar} \F_\P^{b}(\Sigma^{\infty}_+(\mu)) \lrar  \F_\P^{b}(\rL^{b}_{\mathcal{P}}) \; \; , \; \; \rL_{\mathcal{P}/\mathcal{I}_C} \x{\beta}{\lrar} \Sigma^{\infty}_+(\eta)[1] \lrar \rL_{\mathcal{P}}[1] .$$

The map $\alpha$ is described as follows. From Remark \ref{r:necess} we have that $$\F_\P^{b}(\Sigma^{\infty}_+(\mu)) = \Sigma^{\infty} ( \P \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes [1]_\cS) .$$ On other hand, by Lemma \ref{l:relcotanbimod} we have an identification $\F_\P^{b}(\rL^{b}_{\P / \P \circ \P}) \simeq \rL_{\mathcal{P}/\mathcal{I}_C}[1]$ and hence, $\F_\P^{b}(\rL^{b}_{\P / \P \circ \P})[-1]$ is identified to $\rL_{\mathcal{P}/\mathcal{I}_C}$. By Lemma \ref{l:twomodels} we have two models for $\rL_{\mathcal{P}/\mathcal{I}_C}$ given by $\Sigma^{\infty}(\R \, \underset{\I_C}{\bigsqcup} \, \P)$ and $\Sigma^{\infty}(\R' \, \underset{\I_C}{\bigsqcup} \, \P)$.  We use the first one as a model for $\F_\P^{b}(\rL^{b}_{\P / \P \circ \P})[-1]$, i.e.,
$$ \F_\P^{b}(\rL^{b}_{\P / \P \circ \P})[-1] = \Sigma^{\infty}(\R \, \underset{\I_C}{\bigsqcup} \, \P) = \Sigma^{\infty} \left ( (\P \, \bigsqcup \, \P) \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes [1]_\cS \right) .$$
Under these identifications, $\alpha$ is given by the map of suspension spectra $$ \alpha : \Sigma^{\infty} \left ( (\P \, \bigsqcup \, \P) \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes [1]_\cS \right) \lrar \Sigma^{\infty} \left ( \P \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes [1]_\cS \right ) $$
canonically induced by the fold map $\P \, \bigsqcup \, \P \lrar \P$.

Again by Remark \ref{r:necess}, we have $\Sigma^{\infty}_+(\eta)[1] = \Sigma^{\infty} \left ( \P \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes \mathcal{E} \right )$. Furthermore, when using $\Sigma^{\infty}(\R' \, \underset{\I_C}{\bigsqcup} \, \P) = \Sigma^{\infty} \left ( (\P \, \bigsqcup \, \P) \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes \mathcal{E} \right)$ as a model for $\rL_{\mathcal{P}/\mathcal{I}_C}$, then by construction $\beta$ is given by the map of suspension spectra
$$ \beta : \Sigma^{\infty} \left ( (\P \, \bigsqcup \, \P) \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes \mathcal{E} \right) \lrar \Sigma^{\infty} \left ( \P \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes \mathcal{E} \right) $$
canonically induced by the fold map $\P \, \bigsqcup \, \P \lrar \P$ again.

We now obtain a commutative square in $\T_\P \Op(\mathcal{S})$
$$ \xymatrix{
\Sigma^{\infty} \left ( (\P \, \bigsqcup \, \P) \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes [1]_\cS \right)  \ar^{ \; \; \; \; \; \alpha}[r]\ar_{\theta_{\P/\I_C}}^{\simeq}[d] & \Sigma^{\infty} \left ( \P \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes [1]_\cS \right ) \ar_{\simeq}^{\eta^{op}_!(\theta_{\I_C})}[d] \\
\Sigma^{\infty} \left ( (\P \, \bigsqcup \, \P) \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes \mathcal{E} \right) \ar_{ \; \; \; \; \beta}[r] & \Sigma^{\infty} \left ( \P \, \underset{\I_C \sqcup \I_C}{\bigsqcup} \, \I_C \otimes \mathcal{E} \right) \\
in which the left and right vertical maps are the weak equivalences of lemmas \ref{l:twomodels} and \ref{l:muandeta}, respectively. It hence induces a weak equivalence between homotopy cofibers of the maps $\alpha$ and $\beta$. We thus obtain a natural weak equivalence $\theta_\P : \rL^{b}_{\mathcal{P}} \lrarsimeq \rL_{\mathcal{P}}[1]$ in $\T_\P \Op(\mathcal{S})$ as expected.
\end{proof}

In the next step, we will describe the cotangent complex $^b_𝒫 ∈_(𝒫)$. We will need the following lemma.

\begin{lem}\label{bimodules&leftmodules} The forgetful functor $\sU:\BMod(\mathcal{P}) \lrar \LMod(\mathcal{P})$ is a left Quillen functor provided that $\mathcal{P}$ is $\Sigma$-cofibrant.
\end{lem}
\begin{proof} The functor $\sU$ is given by forgetting the right $\P$-action, which is linear. It implies that $\sU$ preserves colimits. By the adjoint functor theorem and by the combinatoriality of $\cS$, $\sU$ is indeed a left adjoint.

Since $\sU$ preserves weak equivalences, the proof will be completed after showing that it preserves cofibrations. To this end, we first prove that every cofibration in $\RMod(\mathcal{P})$ has underlying map in $\Coll_C(\cS)$ being a cofibration as well. Observe that the model structure on $\RMod(\mathcal{P})$ admits a set of generating cofibrations given by $ \{i \circ \P : M \circ \P \lrar N \circ \P \}_i $ where  $i:M\rar N$ ranges over the set of those of $\Coll_C(\cS)$. Since the forgetful functor $\RMod(\mathcal{P}) \lrar \Coll_C(\cS)$ is a left adjoint, it suffices to show that each map $i \circ \P : M \circ \P \lrar N \circ \P$ is a cofibration in $\Coll_C(\cS)$. Let $\emptyset_C$ denote the initial $C$-collection, which agrees with $\emptyset_\cS$ on every level. Then, factor the map $i \circ \P$ as $$M \circ \P \lrar N \circ \emptyset_C \underset{M \circ \emptyset_C}{\bigsqcup} M \circ \P \lrar N \circ \P .$$ The map $i \circ \emptyset_C : M \circ \emptyset_C \lrar N \circ \emptyset_C$ agrees with the component of $i$ on level 0, it is hence a cofibration. So the first map of the above composition is a cofibration. The second map is also a cofibration by [[12],  Lemma 11.5.1], along with the $\Sigma$-cofibrancy of $\P$. Thus, $i \circ \P$ is indeed a cofibration.

It can be shown that the model structure on $\BMod(\mathcal{P})$ admits a set of generating cofibrations given by $\{\P\circ j : \P\circ K \lrar \P\circ L \}_j $ where $j:K\rar L$ ranges over the set of those of $\RMod(\mathcal{P})$. It therefore suffices to show that each map $\P\circ j:\P\circ K\lrar \P\circ L$ is a cofibration in $\LMod(\mathcal{P})$. This is now clear since  $\mathcal{P}\circ(-)$ is the free left $\P$-module functor $\Coll_C(\cS) \lrar \LMod(\mathcal{P})$, which is a left Quillen functor, and since $j$ is a cofibration in $\Coll_C(\cS)$ as indicated above.
\end{proof}

Suppose that $$ is fibrant and $Σ$-cofibrant. Properly, $^b_𝒫$ is given by $Σ^∞(𝒫   b   𝒫^)$ with $𝒫^$ being a cofibrant resolution of $$ in $(𝒫)$. But the map $𝒫   b   𝒫^ 𝒫   b   𝒫$ is a weak equivalence of $$-bimodules due to Lemma \ref{bimodules&leftmodules}, so we will exhibit $Σ^∞(𝒫   b   𝒫)$ as a model for $^b_𝒫$. According to [[8], Corollary 2.3.3], $^b_𝒫=Σ^∞(𝒫   b   )$ admits a \textbf{suspension spectrum replacement} simply given by fixing $𝒫   b   $ as its value at the bidegree $(0,0)$, and hence the value at the bidegree $(n,n)$ is given by $Σ^n(𝒫   b   )$ the $n$-suspension of $𝒫   b    $ in $()_//$. So $^b_𝒫$ is fully determined just by describing the $$-bimodule  $Σ^n(𝒫   b   )$ for every $n⩾0$.

\begin{notns} For each $n \geqslant 0$, we denote by $\sS^{n}:=\Sigma^{n}(1_\mathcal{S}\sqcup 1_\mathcal{S})\in\cS$ with the suspension $\Sigma(-)$ computed in $\mathcal{S}_{1_\mathcal{S}//1_\mathcal{S}}$, and refer to $\sS^{n}$ as the \textbf{pointed} $\textbf{n}$\textbf{-sphere} in $\cS$. Furthermore, we will write $\sS_C^{n}$ standing for the $C$-collection which has $\sS_C^{n}(c;c)=\sS^{n}$ for every $c\in C$ and agrees with $\emptyset_\cS$ on the other levels.
\end{notns}

\begin{comp}\label{computation1} By Lemma~\ref{bimodules&leftmodules}, the underlying left $\mathcal{P}$-module of $\Sigma^{n}(\mathcal{P} \, \underset{b}{\bigsqcup} \,  \mathcal{P})$ is nothing but
$\Sigma^{n}(\mathcal{P}  \, \underset{l}{\bigsqcup} \,  \mathcal{P}) \in \LMod(\mathcal{P})_{\mathcal{P}//\mathcal{P}}$, in which $``\underset{l}{\bigsqcup}$'' refers to the coproduct operation in $\LMod(\mathcal{P})$. The good thing is that $\P$ is free (generated by $\I_C$) as a left module over itself. Thanks to this, we may compute $\Sigma^{n}(\mathcal{P} \, \underset{l}{\bigsqcup} \,  \mathcal{P})$ as follows. First, note that $\mathcal{P} \, \underset{l}{\bigsqcup} \,  \mathcal{P} \in \LMod(\mathcal{P})$ is isomorphic to $\mathcal{P} \circ \sS_C^{0}$ the free left $\P$-module generated by $\sS_C^{0}$. We have further that
$$ \Sigma(\mathcal{P} \, \underset{l}{\bigsqcup} \,  \mathcal{P}) \simeq \mathcal{P}\circ \left( \underset{\sS_C^{0}}{\mathcal{I}_C\;\overset{\h}{\bigsqcup}\;\mathcal{I}_C} \right) \simeq \mathcal{P}\circ \sS_C^{1}.$$
Inductively, we find that $\Sigma^{n}(\mathcal{P} \, \underset{l}{\bigsqcup} \,  \mathcal{P}) \simeq \mathcal{P}\circ \sS_C^{n}$ the free left $\mathcal{P}$-module generated  by $\sS_C^{n}$. In particular, for each $C$-sequence $\overline{c}:=(c_1,\cdots,c_m;c)$, we find that
$$ \Sigma^{n}(\mathcal{P} \, \underset{l}{\bigsqcup} \,  \mathcal{P})\,(\overline{c})\simeq\mathcal{P}\circ \sS_C^{n}\,(\overline{c})=\mathcal{P}(\overline{c})\otimes (\sS^{n})^{\otimes m}.$$
\end{comp}

\begin{notns}\label{notn:LL_P} We denote by $\widetilde{\rL}_\P\in \T_\P\IbMod(\P)$ the derived image of $\rL^{b}_\P\in \T_\P\BMod(\P)$ under the right Quillen equivalence   $\T_\P\BMod(\P) \lrarsimeq \T_\P\IbMod(\P)$ (cf. Theorem \ref{t:mainoptangent}). Furthermore, recall that when $\mathcal{S}$ is in addition stable containing a strict zero object 0,  we have a sequence of right Quillen equivalences
$$ \T_\P\IbMod(\P) \overset{\Omega^{\infty}}{\underset{\simeq}{\longrightarrow}} \IbMod(\mathcal{P})_{\mathcal{P}//\mathcal{P}} \overset{\ker}{\underset{\simeq}{\longrightarrow}} \IbMod(\mathcal{P}) $$
(cf. Theorem \ref{t:mainoptangent'}). In this situation, we will denote by $\ovl{\rL}_\P := \mathbb{R}(\ker\circ \, \Omega^{\infty})(\widetilde{\rL}_\P)$.
\end{notns}

\begin{comp}\phantomsection\label{r:ifSstable} Let us compute $\widetilde{\rL}_\P$ and $\ovl{\rL}_\P$.

\noindent(1) It is not difficult to show that the Quillen adjunction $\adjunction*{}{\IbMod(\mathcal{P})_{\P//\P}}{\BMod(\mathcal{P})_{\P//\P}}{}$ is differentiable (see Definition~\ref{dfndifferentiable}). The [[8], Corollary 2.4.8] hence shows that the right Quillen equivalence $\T_\P\BMod(\P) \lrarsimeq \T_\P\IbMod(\P)$ simply sends $\rL^{b}_{\mathcal{P}}$ (which is now identified to its suspension spectrum replacement)  to its underlying prespectrum of infinitesimal $\P$-bimodules.

\noindent(2) Suppose that $\cS$ is in addition stable containing a strict zero object 0. Let us compute $\ovl{\rL}_\P = \mathbb{R}(\ker\circ \, \Omega^{\infty})(\widetilde{\rL}_\P)$. By [[8], Remark 2.4.7], we have $\mathbb{R}\Omega^{\infty}(\widetilde{\rL}_\P)\simeq \hocolim_n\Omega^{n}(\widetilde{\rL}_\P)_{n,n}$. Now, by Computations~\ref{computation1} we find that
$$\mathbb{R}(\ker\circ \, \Omega^{\infty})(\widetilde{\rL}_\P)\simeq ( \hocolim_n\Omega^{n}(\mathcal{P}\circ \sS_C^{n}) ) \overset{\h}{\times_\P}  0 \simeq \hocolim_n\Omega^{n}  [\,(\mathcal{P}\circ \sS_C^{n})  \overset{\h}{\times_\P}  0 )\,] .$$
More explicitly, for each $\overline{c}=(c_1,\cdots,c_m;c)$ we find that
$$ \mathbb{R}(\ker\circ \, \Omega^{\infty})(\widetilde{\rL}_\P)\,(\overline{c})\simeq \hocolim_n\Omega^{n} [\,(\mathcal{P}(\overline{c})\otimes (\sS^{n})^{\otimes m})  \overset{\h}{\times_{\P(\overline{c})}}  0\,] \simeq \P(\ovl{c})\otimes \hocolim_n\Omega^{n} [\, (\sS^{n})^{\otimes m}  \overset{\h}{\times_{1_\mathcal{S}}}  0\,]  $$
in which the last desuspension $\Omega(-)$ is now computed in $\IbMod(\mathcal{P})$. (In the above formula, the second weak equivalence is because of the fact that homotopy pullbacks in $\cS$ are also homotopy pushouts and that the functor $\mathcal{P}(\overline{c})\otimes (-)$ preserves colimits).

\end{comp}

\begin{cor}\label{conclusionCotangentCplx} Suppose  that $\mathcal{S}$ additionally satisfies the condition (S8) \ref{S8} and that $\P$ is fibrant and $\Sigma$-cofibrant. Then the  right Quillen equivalence $$\U^{ib}_{\P}:\T_\P \Op(\mathcal{S}) \lrarsimeq \T_\P\IbMod(\P)$$ identifies the cotangent complex $\rL_\P$ to $\widetilde{\rL}_\P[-1]\in \T_\P\IbMod(\P)$ in which $\widetilde{\rL}_\P$ is the prespectrum with $(\widetilde{\rL}_\P)_{n,n} = \P\circ \sS_C^{n}$ for $n\geq0$. When $\mathcal{S}$ is in addition stable containing a strict zero object 0 then under the right Quillen equivalence $$ \T_\P \Op(\mathcal{S}) \lrarsimeq \IbMod(\mathcal{P}) ,$$ the cotangent complex $\rL_\P$ is identified to $\ovl{\rL}_\P[-1]$ with $\ovl{\rL}_\P\in \IbMod(\mathcal{P})$ being given by
$$ \ovl{\rL}_\P(\overline{c}) = \P(\ovl{c})\otimes \hocolim_n\Omega^{n} [\, (\sS^{n})^{\otimes m}  \times_{1_\mathcal{S}}^{\h}  0\,] $$
for each $C$-sequence $\overline{c} :=(c_1,\cdots,c_m;c)$.
\end{cor}
\begin{proof} By Proposition~\ref{cotantgentcplxOperadandBimodule} (and by Notations \ref{notn:LL_P}), we get that $\RR\U^{ib}_{\P}(\rL_\P) \simeq \widetilde{\rL}_\P[-1]$. By Computations \ref{r:ifSstable}(1), $\widetilde{\rL}_\P$ agrees with $\rL^{b}_{\mathcal{P}}$ on each level. The description of the latter is included in Computations \ref{computation1}.

In the case where $\cS$ is in addition stable containing a strict zero object, the claim follows just by combining the above paragraph with Computations \ref{r:ifSstable}(2).
\end{proof}

By the definition of Quillen cohomology group~\ref{d:Qcohom}, we give the following conclusion.

\begin{thm}\label{c:quillencohom} Suppose we are given a fibrant object $M\in \T_\P\IbMod(\P)$. Under the same assumptions as in Proposition \ref{conclusionCotangentCplx}, the $n$'th Quillen cohomology group of $\P$ with coefficients in $M$ is formulated as
$$ \sH^{n}_Q(\P,M) \cong \pi_0\Map^{\h}_{\T_\P\IbMod(\P)}(\widetilde{\rL}_\P[-1],M[n]) \cong \pi_0\Map^{\h}_{\T_\P\IbMod(\P)}(\widetilde{\rL}_\P,M[n+1]).$$
Furthermore, suppose that $\cS$ is in addition stable containing a strict zero object 0. For a given fibrant object $M\in \IbMod(\P)$, the $n$'th Quillen cohomology group of $\P$ with coefficients in $M$ is formulated as
$$ \sH^{n}_Q(\P,M) \cong \pi_0\Map^{\h}_{\IbMod(\P)}(\ovl{\rL}_\P[-1],M[n]) \cong \pi_0\Map^{\h}_{\IbMod(\P)}(\ovl{\rL}_\P,M[n+1]).$$
\end{thm}

\smallskip

\subsection{Long exact sequence relating Quillen cohomology and reduced Quillen cohomology}\label{sub:longexact}

\smallskip

The unit map $η: _C $ gives rise to the Quillen adjunctions $$\eta_!^{ib} : \adjunction*{}{\mathcal{T}_{\mathcal{I}_C}\IbMod(\mathcal{I}_C)}{\T_\P \IbMod(\mathcal{P})}{} : \eta^{*}_{ib} \;\;\;\; , \;\;\;\; \eta_!^{op} : \adjunction*{}{\mathcal{T}_{\mathcal{I}_C}\Op(\mathcal{S})}{\mathcal{T}_\mathcal{P}\Op(\mathcal{S})}{} : \eta^{*}_{op}.$$ Moreover, there is a commutative diagram of Quillen adjunctions of the form
\begin{center}
\begin{tikzcd}[row sep=3.5em, column sep =3.5em]
\mathcal{T}_{\mathcal{I}_C}\IbMod(\mathcal{I}_C) \arrow[r, shift left=1.5, "\mathcal{F}_{\mathcal{I}_C}^{ib}"] \arrow[d, shift right=1.5, "\eta_!^{ib}"'] & \mathcal{T}_{\mathcal{I}_C}\Op(\mathcal{S}) \arrow[l, shift left=1.5, "\perp"', "\mathcal{U}_{\mathcal{I}_C}^{ib}"] \arrow[d, shift left=1.5, "\eta_!^{op}"]  & \\
\T_\P \IbMod(\mathcal{P}) \arrow[r, shift right=1.5, "\mathcal{F}^{ib}_\P"'] \arrow[u, shift right=1.5, "\dashv", "\eta^{*}_{ib}"'] & \mathcal{T}_\mathcal{P}\Op(\mathcal{S})  \arrow[l, shift right=1.5, "\downvdash", "\mathcal{U}^{ib}_\P"'] \arrow[u, shift left=1.5, "\vdash"', "\eta^{*}_{op}"] &
\end{tikzcd}
\end{center}

The following is an analogue of [[9], Corollary 3.2.9].
\begin{lem}\label{cofibersequence1} Suppose that $\cS$ additionally satisfies the condition (S8) \ref{S8} and that  $\P$ is fibrant and $\Sigma$-cofibrant. There is a (homotopy) cofiber sequence in $\T_\P\IbMod(\P)$ of the form
\begin{equation}\label{eq:cofib2}
\RR \U^{ib}_{\P,C}(\rL_\P^{\red}) \lrar \LL\eta_!^{ib}\,(\widetilde{\rL}_{\I_C}) \lrar \widetilde{\rL}_{\P}
\end{equation}
where $\U^{ib}_{\P,C}$ is the right Quillen equivalence $\U^{ib}_{\P,C} : \T_\P \Op_C(\mathcal{S}) \lrarsimeq \T_\P\IbMod(\P)$ appearing in Theorem \ref{t:mainoptangent}.
\end{lem}
\begin{proof} Corollary \ref{conclusionCotangentCplx} proves the existence of weak equivalences $\widetilde{\theta}_{\I_C} : \widetilde{\rL}_{\I_C}[-1]\overset{\simeq}{\lrar}\mathbb{R}\mathcal{U}_{\mathcal{I}_C}^{ib}(\rL_{\mathcal{I}_C})$ and $ \widetilde{\theta}_\P:\widetilde{\rL}_{\P}[-1]\overset{\simeq}{\lrar}\mathbb{R}\mathcal{U}^{ib}_\P(\rL_{\mathcal{P}})$ in $\mathcal{T}_{\mathcal{I}_C}\IbMod(\mathcal{I}_C)$ and $\mathcal{T}_{\P}\IbMod(\P)$, respectively. Applying $\LL\eta_!^{op}$ to $\widetilde{\theta}_{\I_C}^{ad}:\LL\mathcal{F}_{\mathcal{I}_C}^{ib}\,(\widetilde{\rL}_{\I_C})\lrarsimeq \rL_{\mathcal{I}_C}[1]$ (i.e., the adjoint of $\widetilde{\theta}_{\I_C}$), and taking then the adjoint of the resultant, we obtain a weak equivalence in $\T_\P\IbMod(\P)$ of the form $\LL\eta_!^{ib}\,(\widetilde{\rL}_{\I_C})\overset{\simeq}{\lrar}\RR\mathcal{U}^{ib}_\P\,\LL\eta_!^{op} (\rL_{\mathcal{I}_C}[1])$.

On other hand, by the definition of relative cotangent complex, there is a cofiber sequence in $\tpop$ of the form $$ \rL_{\P/\I_C} \lrar \LL\eta_!^{op}(\rL_{\I_C})[1] \lrar \rL_\P[1]  .$$
By applying $\mathbb{R}\mathcal{U}^{ib}_\P$ to the latter and by the first paragraph, we get a cofiber sequence in $\T_\P\IbMod(\P)$:
$$  \mathbb{R}\mathcal{U}^{ib}_\P(\rL_{\P/\I_C}) \lrar \LL\eta_!^{ib}\,(\widetilde{\rL}_{\I_C})\lrar \widetilde{\rL}_{\P} .$$
Now, note that the functor $\mathcal{U}^{ib}_\P$ is the same as the composition $ \U^{ib}_{\P,C}\circ\mathcal{R}^{\Sp}_\P$. Lemma~\ref{CotangentCplxOpC&OP} hence shows that  there is a weak equivalence $\RR \U^{ib}_{\P,C}(\rL_\P^{\red}) \lrarsimeq \mathbb{R}\mathcal{U}^{ib}_\P(\rL_{\P/\I_C})$ in $\T_\P\IbMod(\P)$. So we get the desired cofiber sequence.
\end{proof}

We end this section with the following theorem.

\begin{thm}\label{c:longexseq1} Suppose that $\cS$ additionally satisfies the condition (S8) \ref{S8} and that $\mathcal{P}$ is fibrant and $\Sigma$-cofibrant. Given a fibrant object $M\in \T_\P\IbMod(\P)$, there is a long exact sequence of abelian groups of the form
$$ \cdots \longrightarrow \sH^{n-1}_Q (\mathcal{P},M) \lrar  \sH^{n}_{Q,r} (\mathcal{P},M) \lrar \sH^{n}_{Q,\red} (\mathcal{P},M) \lrar \sH^{n}_Q (\mathcal{P},M) \lrar \sH^{n+1}_{Q,r} (\mathcal{P},M)  \lrar \cdots $$
where $\sH^{\bullet}_{Q,r} (\mathcal{P},-)$ refers to Quillen cohomology group of $\P$ when regarded as a right module over itself, while $\sH^{\bullet}_Q (\mathcal{P},-)$ refers to Quillen cohomology group of $\P$ and $\sH^{\bullet}_{Q,\red} (\mathcal{P},-)$ refers to reduced Quillen cohomology group of $\P$ (cf. Conventions \ref{conv:reducedQcohom}).
\end{thm}
\begin{proof} The cofiber sequence of Lemma \ref{cofibersequence1} induces a fiber sequence of derived mapping spaces:
$$\Map^{\h}_{\T_\P\IbMod(\P)}(\widetilde{\rL}_{\P} , M) \longrightarrow \Map^{\h}_{\T_\P\IbMod(\P)}(\LL\eta_!^{ib}\,(\widetilde{\rL}_{\I_C}) , M) \longrightarrow \Map^{\h}_{\T_\P\IbMod(\P)}(\RR \U^{ib}_{\P,C}(\rL_\P^{\red}) , M) .$$
In this sequence, by notation $\RR \U^{ib}_{\P,C}(\rL_\P^{\red})$ classifies the reduced Quillen cohomology of $\P$, while $\widetilde{\rL}_{\P}$ classifies the Quillen cohomology of $\P$, by Theorem \ref{c:quillencohom}. That fiber sequence will hence give rise to the desired long exact sequence after having that $\LL\eta_!^{ib}\,(\widetilde{\rL}_{\I_C})$ classifies the Quillen cohomology of $\P$ when regarded as a right module over itself. To see this, we first consider the Quillen adjunction $\adjunction*{}{\T_\P\RMod(\P)}{\T_\P\IbMod(\P)}{}$ which is induced by the free-forgetful adjunction $\adjunction*{}{\RMod(\P)}{\IbMod(\P)}{}$. We denote by $\rL^{r}_\P \in \T_\P\RMod(\P)$ the cotangent complex of $\P$ when regarded as a right module over itself. It therefore suffices to prove that the derived image of $\rL^{r}_\P$ in $\T_\P\IbMod(\P)$ is weakly equivalent to $\LL\eta_!^{ib}\,(\widetilde{\rL}_{\I_C})$. For this last claim, observe first that $\eta_!^{ib}$ is the same as the functor $\T_{\I_C}\Coll_C(\cS) \lrar \T_\P\IbMod(\P)$ induced by the free functor $\Coll_C(\cS) \lrar \IbMod(\P)$. Moreover, under the identification $\mathcal{T}_{\mathcal{I}_C}\IbMod(\mathcal{I}_C) \simeq \T_{\I_C}\Coll_C(\cS)$, the object $\widetilde{\rL}_{\I_C}$ is nothing but the cotangent complex of $\mathcal{I}_C$ when regarded as an object of $\Coll_C(\cS)$, which has the derived image through the left Quillen functor $\T_{\I_C}\Coll_C(\cS) \lrar \T_\P\RMod(\P)$ being $\rL^{r}_\P$ obviously. The proof is hence completed.
\end{proof}

\begin{rem} When considering $\mathcal{P}$ as a right module over itself, it is free generated by $\I_C$. Thus, we have a canonical isomorphism
$$ \sH^{n}_{Q,r} (\mathcal{P},M) \cong \sH^{n}_{Q,col} (\I_C,\ovl{M}) ,$$
where the right hand side is the $n$'th Quillen cohomology group of $\I_C$, when regarded as a $C$-collection, with coefficients in the derived image of $M$ in $\T_{\I_C}\Coll_C(\cS)$. When $\cS$ is further stable containing a strict zero object 0 (and hence, so is the category $\Coll_C(\cS)$), we have a sequence of right Quillen equivalences
$$ \T_{\I_C}\Coll_C(\cS) \x{\Omega^{\infty}}{\lrarsimequ} \Coll_C(\cS)_{\I_C//\I_C} \x{\ker}{\lrarsimequ} \Coll_C(\cS) $$
(cf. Theorem \ref{t:mainoptangent'}) and moreover, under this composed right Quillen equivalence, the cotangent complex of $\I_C$ is identified to itself $\I_C$ (cf. [[10],  Corollary 2.2.4]). Therefore, in this situation we have that
$$ \sH^{n}_{Q,col} (\I_C,\ovl{M}) \cong  \pi_0\Map^{\h}_{\Coll_C(\cS)}(\I_C,\RR\ker\Omega^{\infty}(\Sigma^{n}\ovl{M})) \cong \bigsqcap_{c\in C} \pi_0\Map^{\h}_{\cS}(1_\cS,\Sigma^{n}(\ovl{M}(c;c)\times_{1_\cS}0))  .$$
\end{rem}

\smallskip

\section{Quillen cohomology of simplicial operads}\label{s:simplicialoperad}

\smallskip

Simplicial operads are precisely operads enriched over the Cartesian monoidal category of simplicial sets, $_Δ$. This category comes equipped with the standard (Kan-Quillen) model structure and then, satisfies the conditions of Conventions \ref{convention} and also the extra condition (S8) \ref{S8} (cf. Example \ref{ex:basecategories} and Proposition \ref{p:splcsetS9}). We therefore inherit the results of $$\ref{s:cotangentcplx} for the work of this section.

In the first subsection, we revisit the \textbf{straightening} and \textbf{unstraightening} constructions (in unmarked case), according to [[4], \S 2.2.1]. Given a simplicial (co)presheaf  $$ over a simplicial category $$, the unstraightening of $$ is in particular a simplicial set over $$. As the first step, we will focus on describing the simplices of the unstraightening of $$ and furthermore, giving convenient models for its \textbf{spaces of (right) left morphisms}.

According to the work of Y. Harpaz, J. Nuiten and M. Prasma ([9]), the cotangent complex of an $∞$-category (or a fibrant simplicial category) can be represented as a spectrum valued functor on its \textbf{twisted arrow $\infty$-category} (see Theorem \ref{t:Qcohominftycat}). The construction of twisted arrow $∞$-categories (of $∞$-categories) $(-) : _∞_∞$ was originally introduced by Lurie [[3], $$ 5.2]. For a given fibrant simplicial category $$, the twisted arrow $∞$-category of $$ is simply defined to be $():=()$ with $$ being the \textbf{simplicial nerve} of $$. In the second subsection, we extend the latter to the construction of twisted arrow $∞$-categories of (fibrant) simplicial operads. For a fibrant simplicial operad $$, the twisted arrow $∞$-category $()$ is by definition the unstraightening of the simplicial copresheaf $: Ib^𝒫 _Δ$, which encodes the data of $$ as an infinitesimal bimodule over itself (see $$\ref{s:algebrasencodingmodules}). The results obtained from the first subsection provide us with a comprehensive view on the structure of $()$, on both the simplicial and $∞$-categorical structures.

In the last subsection, for the main purpose we will show that the cotangent complex of a simplicial operad can be represented as a spectrum valued functor on its twisted arrow $∞$-category.

\subsection{Unstraightening of simplicial (co)presheaves}

\smallskip

We denote by $ℭ[-] : _Δ(_Δ)$ the \textbf{rigidification functor}, which is left adjoint to the simplicial nerve functor $$. Recall by definition that $ℭ[Δ^n]$ is the simplicial category whose set of objects is $[n] = {0,⋯,n}$ and whose mapping spaces are defined as $_ℭ[Δ^n](i,j) := _i,j$ the nerve of the poset $$\rP_{i,j} = \{A \, | \, \{i,j\} \subseteq A \subseteq [i,j] \} ,$$
with $[i,j] := {i,i+1,⋯,j}$. (In particular, $_ℭ[Δ^n](i,j) = ∅$ when $i>j$). The composition in $ℭ[Δ^n]$ is induced by the maps  $_i,j ×_j,k _i,k$ taking $(A,B)$ to $A ∪B$.

Depending if one wants to  $“$unstraighten'' simplicial presheaves or copresheaves, one would need to use \textbf{contravariant} or \textbf{covariant unstraightening} functor, respectively.  Suppose we are given a simplicial set $S$ and a simplicial category $$. Let $ϕ: ℭ[S] $ be a simplicial functor. For each simplicial set $X$ over $S$, one takes two simplicial categories
$$\M_X^{\triangleleft} := \mathfrak{C}[X^{\triangleleft}]  \underset{\mathfrak{C}[X]}{\bigsqcup} \C \; \; \; , \; \; \; \M_X^{\triangleright} := \mathfrak{C}[X^{\triangleright}]  \underset{\mathfrak{C}[X]}{\bigsqcup} \C$$
where $X^◃$ and $X^▹$ are respectively the left and right cones of $X$. We will always use the letter $“v$'' standing for the cone point, for both left and right cones.

The \textbf{covariant straightening functor} associated to $ϕ$, written as
$$ \St_{\phi}^{\triangleleft} : (\Set_\Delta)_{/S} \lrar \Fun(\C , \Set_\Delta) ,$$
is defined by sending each simplicial set $X$ over $S$ to the functor $__X^◃(v,-) : _Δ$. The functor $_ϕ^◃$ admits a right adjoint, the \textbf{covariant unstraightening functor} associated to $ϕ$, denoted by $_ϕ^◃$. In fact, the adjunction $_ϕ^◃ ⊣_ϕ^◃$ forms a Quillen adjunction when one endows the category $(, _Δ)$ with the projective model structure and the category $(_Δ)_/S$ with the \textbf{covariant model structure}. Moreover, this adjunction is a Quillen equivalence as long as $ϕ$ is a weak equivalence of simplicial categories. (See [4], Chapter 3 for more details).

Dually, the \textbf{contravariant straightening functor} associated to $ϕ$, written as
$$ \St_{\phi}^{\triangleright} : (\Set_\Delta)_{/S} \lrar \Fun(\C^{\op} , \Set_\Delta) ,$$
is defined by sending each simplicial set $X$ over $S$ to the functor $__X^▹(-,v) : ^ _Δ$. The functor $_ϕ^▹$ admits a right adjoint, the \textbf{contravariant unstraightening functor} associated to $ϕ$, denoted by $_ϕ^▹$. The adjunction $_ϕ^▹ ⊣_ϕ^▹$ forms a Quillen adjunction when one endows the category $(^ , _Δ)$ with the projective model structure and the category $(_Δ)_/S$ with the \textbf{contravariant model structure} and moreover, becomes a Quillen equivalence if $ϕ$ is a weak equivalence.

\begin{notns}\phantomsection\label{no:unstoverS}
\begin{enumerate}
\item We are almost concentrated in the case where $S = \sN \C$ and $\phi = \varepsilon_\C : \mathfrak{C}[\sN \C] \lrar \C$ the counit map of $\mathfrak{C}[-] \dashv \sN$. In this case, the corresponding covariant (resp. contravariant) straightening-unstraightening adjunction will be denoted by $\St_{\C}^{\triangleleft} \dashv \Un_{\C}^{\triangleleft}$ (resp. $\St_{\C}^{\triangleright} \dashv \Un_{\C}^{\triangleright}$).
\item When $\C = \mathfrak{C}[S]$ and $\phi$ is the identity functor, the corresponding covariant (resp. contravariant) straightening-unstraightening adjunction will be denoted by $\St_{S}^{\triangleleft} \dashv \Un_{S}^{\triangleleft}$ (resp. $\St_{S}^{\triangleright} \dashv \Un_{S}^{\triangleright}$).
\end{enumerate}
\end{notns}

\begin{rem}\label{r:Un_C} Note that $\Un_{\C}^{\triangleleft}$ is the same as the composed functor $$\Fun(\C , \Set_\Delta) \x{\varepsilon_\C^{*}}{\xrightarrow{\hspace*{0.7cm}}} \Fun(\mathfrak{C}[\sN \C] , \Set_\Delta) \x{\Un_{\sN\C}^{\triangleleft}}{\xrightarrow{\hspace*{0.7cm}}} (\Set_\Delta)_{/\sN\C} ,$$ (this follows from [4], 2.2.1.1(2)). The same thing holds for the contravariant case.
\end{rem}

Let $: _Δ$ be a simplicial functor. We wish to understand the structure of $_^◃()$ as a simplicial set over $$. For this, we will follow [[43], \S 58], yet in the opposite convention. For each $n∈$, one establishes a simplicial functor
$$ \mathfrak{D}^{\triangleleft}_{\Delta^{n}} : \mathfrak{C}[\Delta^{n}] \lrar \Set_\Delta $$
given by sending each $i∈[n]$ to $𝔇^◃_Δ^n(i):=^◃(i)$ the nerve of the poset $$\rP^{\triangleleft}(i) := \{A \, | \, \{i\}\subseteq A \subseteq [0,i] \, \}$$ with $[0,i] := {0,1,⋯,i}$. The structure maps of simplicial functor are defined by applying the union operation of subsets in an obvious way. Moreover, for each map $δ: Δ^mΔ^n$, one defines a natural transformation
$$ \mathfrak{D}^{\triangleleft}_{\delta} : \mathfrak{D}^{\triangleleft}_{\Delta^{m}} \lrar \mathfrak{D}^{\triangleleft}_{\Delta^{n}} \circ \mathfrak{C}[\delta] $$
of the simplicial functors $ℭ[Δ^m] _Δ$ given at each $i∈[m]$ by the map $𝔇^◃_δ(i) : 𝔇^◃_Δ^m(i)𝔇^◃_Δ^n(δi)$ which is induced by the map of posets $ ^◃(i) ^◃(δi)   ,   S ↦δ(S) $.

\begin{cons}\label{con:Un_CF} The data of an $n$-simplex of $\Un_{\C}^{\triangleleft}(\F)$ consists of

$\bullet$ an $n$-simplex  $\varphi\in\sN(\C)$, i.e., a functor $\varphi : \mathfrak{C}[\Delta^{n}]\lrar \C$, and

$\bullet$ a natural transformation $t:\mathfrak{D}^{\triangleleft}_{\Delta^{n}}\lrar \F\circ \varphi$ between simplicial functors $\mathfrak{C}[\Delta^{n}] \lrar \Set_\Delta$.

\noindent For each map $\delta : \Delta^{m}\lrar\Delta^{n}$, the corresponding structure map $\Un_{\C}^{\triangleleft}(\F)_n\lrar\Un_{\C}^{\triangleleft}(\F)_m$ is given by sending each pair $(\varphi,t)\in \Un_{\C}^{\triangleleft}(\F)_n$  to the pair
$$ \mathfrak{C}[\Delta^{m}] \x{\mathfrak{C}[\delta]}{\xrightarrow{\hspace*{0.7cm}}} \mathfrak{C}[\Delta^{n}] \x{\varphi}{\xrightarrow{\hspace*{0.7cm}}} \C \, \; , \, \; \mathfrak{D}^{\triangleleft}_{\Delta^{m}} \x{\mathfrak{D}^{\triangleleft}_{\delta}}{\xrightarrow{\hspace*{0.7cm}}} \mathfrak{D}^{\triangleleft}_{\Delta^{n}} \circ \mathfrak{C}[\delta] \x{t\circ \Id}{\xrightarrow{\hspace*{0.7cm}}} \F\circ \varphi \circ \mathfrak{C}[\delta] .$$
\end{cons}

\smallskip

\underline{\textit{Explanations:}} First, argue that giving an $n$-simplex $z ∈_^◃()$ is equivalent to giving a sequence of maps $Δ^n z _^◃() $. Let us denote this composite map by $φ_z : Δ^n $, which is identified to a simplicial functor $φ_z : ℭ[Δ^n]$. Observe now that for each such map $φ_z$ there is a commutative square of right adjoint functors
$$ \xymatrix{
\Fun(\C , \Set_\Delta) \ar^{ \; \; \; \; \; \; \Un_{\C}^{\triangleleft}}[r]\ar_{\varphi_z^{*}}[d] & (\Set_\Delta)_{/\sN\C} \ar^{\varphi_z^{*}}[d] \\
\Fun(\mathfrak{C}[\Delta^{n}],\Set_\Delta) \ar_{ \; \; \; \; \; \; \; \; \; \Un_{\Delta^{n}}^{\triangleleft}}[r] & (\Set_\Delta)_{/\Delta^{n}} \\
} $$
(this follows by combining both two parts of [[4], Proposition 2.2.1.1], along with noting Remark \ref{r:Un_C}). This proves the existence of a Cartesian square of the form
$$ \xymatrix{
\Un^{\triangleleft}_{\Delta^{n}}(\F \circ \varphi_z) \ar[r]\ar[d] & \Delta^{n} \ar[d] \\
\Un_{\C}^{\triangleleft}(\F) \ar[r] & \sN\C \\
} $$
So the data of $z$ is equivalent to that of a map $φ_z : Δ^n $ and a map $$\Delta^{n} \lrar \Un^{\triangleleft}_{\Delta^{n}}(\F \circ \varphi_z)$$ factoring the identity $_Δ^n$. By adjunction, the latter is identified to a natural transformation $^◃_Δ^n(_Δ^n) ∘φ_z$. But, by definition  $^◃_Δ^n(_Δ^n) = _ℭ[(Δ^n)^◃](v,-)$. Hence, it remains to establish for each $n$ an isomorphism  $$ \mathfrak{D}^{\triangleleft}_{\Delta^{n}}  \lrarsimeq \Map_{\mathfrak{C}[(\Delta^{n})^{\triangleleft}]}(v,-)$$
between simplicial functors $ℭ[Δ^n] _Δ$ compatible with simplicial maps $Δ^m Δ^n$. For this, note that under the identification $(Δ^n)^◃ ≅Δ^n+1$, the functor $_ℭ[(Δ^n)^◃](v,-)$ is isomorphic to the functor $[n] ∋i ↦_ℭ[Δ^n+1](0,i+1) = _0,i+1$. For each $i ∈[n]$, we define an isomorphism of posets $$\rP^{\triangleleft}(i) \lrarsimeq \rP_{0,i+1} \; \; , \; \; A \mapsto \{0\} \sqcup (A+1)$$ with $A+1 := {a+1   |   a ∈A }$. This indeed gives us the desired isomorphism $𝔇^◃_Δ^n  _ℭ[(Δ^n)^◃](v,-)$.

\smallskip

\begin{rem}\label{r:vertexedge}  Unwinding definition, each vertex of $\Un_{\C}^{\triangleleft}(\F)$ is the choice of an object $x \in \Ob(\C)$ and a vertex $\mu \in \F(x)$. Let $x,y$ be two objects of $\C$. The data of an edge of $\Un_{\C}^{\triangleleft}(\F)$ from $\mu \in \F(x)$ to $\nu \in \F(y)$ consists of a vertex $\alpha \in \Map_\C(x,y)$ and an edge of $\F(y)$ of the form $t : \nu \lrar \alpha_{*}(\mu)$ where $\alpha_{*} : \F(x) \lrar \F(y)$ is the map induced by the simplicial functor structure of $\F : \C \lrar \Set_\Delta$.
\end{rem}

\begin{com} Intuitively, there seems to be something illogical in the data of an edge of $\Un_{\C}^{\triangleleft}(\F)$ mentioned above. Indeed, if one thinks of the covariant unstraightening functor as something generalizing the classical (covariant) Grothendieck construction then it is natural to require that, in the data of such an edge of $\Un_{\C}^{\triangleleft}(\F)$, the edge $t$ should be in the direction $\alpha_{*}(\mu) \lrar \nu$ instead of $\nu \lrar \alpha_{*}(\mu)$. In fact, this confusion arises from the convention, namely, when one chose the adjunction $\mathfrak{C}[-] \dashv \sN$ to work with, for both covariant and contravariant cases. Let us try to modify this convention slightly. We define a new $``$rigidification functor''  $\mathfrak{C}'[-]$ by taking, for each $n$, $\Map_{\mathfrak{C}'[\Delta^{n}]}(i,j) := \sN \rP_{i,j}^{\op}$, (instead of $\sN \rP_{i,j}$ as in the definition of $\mathfrak{C}[-]$). This leads to a new $``$simplicial functor'' $\sN'$ characterized as the right adjoint to $\mathfrak{C}'[-]$. Now, if instead of $\mathfrak{C}[-] \dashv \sN$, one uses the adjunction $\mathfrak{C}'[-] \dashv \sN'$ in establishing the covariant straightening, then the new covariant unstraightening will satisfy what one expects from the classical Grothendieck construction, i.e., one has the edge $t$ indeed being in the direction $\alpha_{*}(\mu) \lrar \nu$. We believe that when doing the covariant unstraightening, using the adjunction $\mathfrak{C}'[-] \dashv \sN'$ is more natural than the $\mathfrak{C}[-] \dashv \sN$. But if one does not expect a perfect generalization from the classical Grothendieck construction anymore, everything still works certainly.
\end{com}

Let $' : ^ _Δ$ be a simplicial functor. Similarly as in the covariant case, one can get an explicit description of $_^▹(')$ the contravariant unstraightening of $'$. For each $n∈$, one defines a simplicial functor
$$ \mathfrak{D}^{\triangleright}_{\Delta^{n}} : \mathfrak{C}[\Delta^{n}]^{\op} \lrar \Set_\Delta $$
taking each $i∈[n]$ to $𝔇^▹_Δ^n(i):=_n^▹(i)$ the nerve of the poset $$\rP_n^{\triangleright}(i) := \{A \, | \, \{i\}\subseteq A \subseteq [i,n] \, \}$$ with $[i,n] := {i,i+1,⋯,n}$.  For each map $δ: Δ^mΔ^n$, there is also a natural transformation $𝔇^▹_δ : 𝔇^▹_Δ^m 𝔇^▹_Δ^n ∘ℭ[δ]^$ of the simplicial functors $ℭ[Δ^m]^ _Δ$. The structure maps are defined naturally as well as in the covariant case (see Construction \ref{con:Un_CF}).

\begin{cons}\label{con:Un_CF'}  The data of an $n$-simplex of $\Un_{\C}^{\triangleright}(\F')$ consists of

$\bullet$ an $n$-simplex  $\varphi\in\sN(\C)$, i.e., a functor $\varphi : \mathfrak{C}[\Delta^{n}] \lrar \C$, and

$\bullet$ a natural transformation $t:\mathfrak{D}^{\triangleright}_{\Delta^{n}}\lrar \F'\circ \varphi^{\op}$ between simplicial functors $\mathfrak{C}[\Delta^{n}]^{\op} \lrar \Set_\Delta$.

\noindent The simplicial structure maps of $\Un_{\C}^{\triangleright}(\F')$ are induced by the natural transformations $\mathfrak{D}^{\triangleright}_{\delta}$ mentioned above. (See also [[43], \S 58]).
\end{cons}

We now consider the case where $= {*}$ the terminal category. The corresponding covariant (resp. contravariant) straightening-unstraightening adjunction will be then denoted by $_*^◃⊣_*^◃$ (resp. $_*^▹⊣_*^▹$). By definition, these forms the self-adjunctions on $_Δ$
$$ \St_{*}^{\triangleleft} : \adjunction*{}{\Set_\Delta}{\Set_\Delta}{} : \Un_{*}^{\triangleleft} \; \; \; , \; \; \; \St_{*}^{\triangleright} : \adjunction*{}{\Set_\Delta}{\Set_\Delta}{} : \Un_{*}^{\triangleright} .$$
Note that $_*^▹⊣_*^▹$ agrees with the adjunction $|-|_Q^∙ ⊣_Q^∙$ of [[4], \S 2.2.2].

Let  $X$ be a simplicial set. We will need explicit descriptions of both $_*^◃(X)$ and $_*^▹(X)$. Recall that the $n$-cube $(Δ^1)^×{1,⋯,n}$ has total of $n$ \textbf{end faces} (i.e., the faces containing the terminal vertex), including
$$ T_i := (\Delta^{1})^{\times\{1,\cdots,i-1\}} \times \{1\} \times (\Delta^{1})^{\times\{i+1,\cdots,n\}} \subseteq (\Delta^{1})^{\times\{1,\cdots,n\}} $$
for $i=1,⋯,n$. For each $i$, we denote by $$T_i^{\circ} := \{1\} \times \cdots \times \{1\} \times (\Delta^{1})^{\times\{i+1,\cdots,n\}} \subseteq T_i ,$$ which is an  $(n-i)$-subcube of $T_i$.

\begin{dfn} Let $\gamma : (\Delta^{1})^{\times\{1,\cdots,n\}} \lrar X$ be a simplicial map performing an $n$-cube in $X$. Then $\gamma$ is called a \textbf{quasi} $n$-\textbf{simplex} of $X$ if for every $i \in \{1,\cdots,n\}$, the end face of $\gamma$ given by $\restr{\gamma}{T_i}$ the restriction of $\gamma$ to $T_i$ is degenerate on $\restr{\gamma}{T_i^{\circ}}$ the restriction of $\gamma$ to $T_i^{\circ}$.
\end{dfn}

\begin{rem}\label{r:quasisimplex} Each $n$-simplex $\ovl{\delta} : \Delta^{n} \lrar X$ corresponds to a quasi $n$-simplex of $X$ given by the composition
$$ \delta : (\Delta^{1})^{\times\{1,\cdots,n\}} \x{\tau}{\lrar} \Delta^{n} \x{\ovl{\delta}}{\lrar} X $$
where the map $\tau$ is given by the canonical way of collapsing the $n$-cube to a nondegenerate $n$-subsimplex containing the initial vertex. Conversely, each quasi $n$-simplex of $X$ agrees with an $n$-cube in $X$ of such type $\delta$ on boundary. This phenomenon itself suggests us to the terminology $``$quasi simplex''.
\end{rem}

\begin{example} A quasi $3$-simplex of $X$ is a 3-cube in $X$ of the form
\begin{center}
\begin{tikzcd}[row sep=0.8em, column sep = 0.8em]
x \arrow[rr] \arrow[dr,dashed, swap] \arrow[dd,swap] &&
y \arrow[dd] \arrow[dr] \\
& z \arrow[rr] \arrow[dd] &&
z \arrow[dd, "f"] \\
t \arrow[rr,] \arrow[dr] && t \arrow[dr] \\
& t \arrow[rr] && t
\end{tikzcd}
\end{center}
such that the front square is degenerate on the edge $z \x{f}{\lrar}t$ and the bottom square is degenerate on the terminal vertex $t$.
\end{example}

The main interest in this notion is as follows.

\begin{lem}\phantomsection\label{l:quasisimplex}
\begin{enumerate}
\item There is a natural bijection between the $n$-simplices of $\Un_{*}^{\triangleright}(X)$ and the quasi $n$-simplices of $X$.
\item Also, there is a natural bijection between the $n$-simplices of $\Un_{*}^{\triangleleft}(X)$ and the quasi $n$-simplices of $X$.
\end{enumerate}
\end{lem}

To get an explicit proof, we make use of $𝔇^▹_Δ^n(0)=_n^▹(0)$ as a specific model for the $n$-cube $(Δ^1)^×{1,⋯,n}$ (see Construction \ref{con:Un_CF'}). For each $i ∈{1,⋯,n}$, we take the subposets
$$ \{A \in \rP_n^{\triangleright}(0) \, | \,  A \supseteq [0,i] \} \; \subseteq \; \{A \in \rP_n^{\triangleright}(0) \, | \,  A \ni i \} \; \subseteq \; \rP_n^{\triangleright}(0) .$$
Note that the inclusion $  {A ∈_n^▹(0)   |    A ⊇[0,i] }   ⊆    {A ∈_n^▹(0)   |    A ∋i }$  corresponds to the inclusion $T_i^∘ ⊆T_i$ mentioned above. Moreover, there is an obvious identification of posets $$\{A \in \rP_n^{\triangleright}(0) \, | \,  A \supseteq [0,i] \} \, \cong \, \rP_n^{\triangleright}(i) .$$

\begin{proof}[\underline{Proof of Lemma~\ref{l:quasisimplex}}] \; (1) \; By Construction \ref{con:Un_CF'}, an $n$-simplex of $\Un_{*}^{\triangleright}(X)$ is a natural transformation  $t':\mathfrak{D}^{\triangleright}_{\Delta^{n}} \lrar \{X\}$ between simplicial functors $\mathfrak{C}[\Delta^{n}]^{\op} \lrar \Set_\Delta$, with $\{X\}$ being the constant functor on $X$. Unwinding definition, the data of $t'$ consists of the maps $t'(i) : \mathfrak{D}^{\triangleright}_{\Delta^{n}}(i) = \sN \rP_n^{\triangleright}(i) \lrar X$ for $i \in [n]$, satisfying that:

\smallskip

(*) for every pair $(i,j)$ such that $i>j$, the composition
\begin{equation}\label{eq:firstcompose}
\Map_{\mathfrak{C}[\Delta^{n}]^{\op}}(i,j) \times \mathfrak{D}^{\triangleright}_{\Delta^{n}}(i) \x{\mathfrak{D}^{\triangleright}_{\Delta^{n}}}{\xrightarrow{\hspace*{0.7cm}}} \mathfrak{D}^{\triangleright}_{\Delta^{n}}(j) \x{t'(j)}{\xrightarrow{\hspace*{0.7cm}}} X
\end{equation}
agrees with the composition $\Map_{\mathfrak{C}[\Delta^{n}]^{\op}}(i,j) \times \mathfrak{D}^{\triangleright}_{\Delta^{n}}(i) \lrar \mathfrak{D}^{\triangleright}_{\Delta^{n}}(i) \x{t'(i)}{\xrightarrow{\hspace*{0.7cm}}} X$.

\smallskip

Note that the map \eqref{eq:firstcompose} corresponds to the restriction of $t'(j)$ to $\sN \, \{A \in \rP_n^{\triangleright}(j) \, | \, A \ni i \}$. When $j=i-1$, the condition (*) means that $t'(i)$ agrees with the restriction of $t'(i-1)$ to $$\sN \, \{A \in \rP_n^{\triangleright}(i-1) \, | \, A \ni i \} \, \cong \, \sN\rP_n^{\triangleright}(i).$$
Inductively, for every $i\geqslant1$, $t'(i)$ agrees with  the restriction of $t'(0)$ to $$\sN \, \{A \in \rP_n^{\triangleright}(0) \, | \,  A \supseteq [0,i] \} \, \cong \, \sN\rP_n^{\triangleright}(i) .$$ In particular, the data of $t'$ is fully displayed in that of $t'(0)$, which performs an $n$-cube in $X$. Now, fix $j=0$, the condition (*) means that for every $i \in \{1,\cdots,n\}$, the end face of $t'(0)$ given by the restriction $\restr{t'(0)}{\sN \, \{A \in \rP_n^{\triangleright}(0) \, | \,  A \ni i \}}$ is degenerate on $\restr{t'(0)}{\sN \, \{A \in \rP_n^{\triangleright}(0) \, | \,  A \supseteq [0,i] \}}$. This is equivalent to saying that $t'(0)$ is a quasi $n$-simplex of $X$, by the words before the proof.

Conversely, let $t' : \mathfrak{D}^{\triangleright}_{\Delta^{n}}(0)=\sN \rP_n^{\triangleright}(0) \lrar X$ be a quasi $n$-simplex of $X$. For each $i\in [n]$, we take  $t'(i)$ to be the restriction of $t'$ to $\sN \, \{A \in \rP_n^{\triangleright}(0) \, | \,  A \supseteq [0,i] \} \, \cong \, \sN\rP_n^{\triangleright}(i)$. In particular, $t'(0)$ is nothing but $t'$. It can then be verified that $\{t'(i)\}_i$ forms a natural transformation $\mathfrak{D}^{\triangleright}_{\Delta^{n}} \lrar \{X\}$.

The two above paragraphs clearly provide the inverses of each other between the $n$-simplices of $\Un_{*}^{\triangleright}(X)$ and the quasi $n$-simplices of $X$.

\smallskip

(2) \; By the first part, it suffices to prove the existence of a natural bijection between the $n$-simplices of $\Un_{*}^{\triangleleft}(X)$ and those of $\Un_{*}^{\triangleright}(X)$. By Construction \ref{con:Un_CF}, an $n$-simplex of $\Un_{*}^{\triangleleft}(X)$ is a natural transformation  $ t : \mathfrak{D}^{\triangleleft}_{\Delta^{n}} \lrar \{X\}$ between simplicial functors $\mathfrak{C}[\Delta^{n}] \lrar \Set_\Delta$ and therefore, it consists of the maps $t(i) : \mathfrak{D}^{\triangleleft}_{\Delta^{n}}(i) = \sN \rP^{\triangleleft}(i) \lrar X  \; (i \in [n])$
subject to the essential naturality with respect to every space of morphisms in $\mathfrak{C}[\Delta^{n}]$ (similarly as the condition (*) above).

For each $i\in [n]$, there is an isomorphism of posets $\rP_n^{\triangleright}(i) \lrarsimeq \rP^{\triangleleft}(n-i)$ taking $A\in \rP_n^{\triangleright}(i)$ to $ n-A := \{n-a \, | \, a \in A \}$. We then take $t'(i)$ to be the composition $\sN \rP_n^{\triangleright}(i) \cong \sN\rP^{\triangleleft}(n-i) \x{t(n-i)}{\xrightarrow{\hspace*{0.8cm}}} X$. It can then be verified that $t' = \{t'(i)\}_i$ forms a natural transformation $t' : \mathfrak{D}^{\triangleright}_{\Delta^{n}} \lrar \{X\}$  and hence, performs an $n$-simplex of $\Un_{*}^{\triangleright}(X)$. Moreover, the assignment $t \mapsto t'$ indeed forms a natural bijection.
\end{proof}

\begin{rem} Though the assignment $t \mapsto t'$ gives an isomorphism between $\Un_{*}^{\triangleleft}(X)$ and $\Un_{*}^{\triangleright}(X)$ on point-set level, it is not compatible with simplicial structures. Instead of that, this turns out to form a natural isomorphism $\Un_{*}^{\triangleleft}(X) \cong \Un_{*}^{\triangleright}(X)^{\op}$.
\end{rem}

\begin{rem}\label{r:XKan} As mentioned in Remark \ref{r:quasisimplex}, each $n$-simplex of $X$ is associated to a quasi $n$-simplex. This determines a natural map $X \lrar \Un_{*}^{\triangleright}(X)$, which is in fact a weak equivalence as long as $X$ is a Kan complex (see [[4], 2.2.2.7, 2.2.2.8]). Combined with the above remark, we get a natural weak equivalence $X^{\op} \lrarsimeq \Un_{*}^{\triangleleft}(X)$ when $X$ is Kan. In other words, while the (derived) contravariant unstraightening $\Un_{*}^{\triangleright}$ is weakly equivalent to the identity functor, the covariant one $\Un_{*}^{\triangleleft}$ is weakly equivalent to the opposite functor $X \mapsto X^{\op}$.
\end{rem}

Given a simplicial category $$ and a simplicial functor $: _Δ$, we turn back to consider the covariant unstraightening of $$. We will need the following in the last subsection.

\begin{rem}\label{r:fiberkan} As in [[4], Remark 2.2.2.11], for a given object $x \in \C$, there is a canonical isomorphism $$\Un_{\C}^{\triangleleft}(\F) \times_{\sN\C} \{x\} \cong \Un_{*}^{\triangleleft}(\F(x)) .$$ This follows by the compatibility of the unstraightening functor with taking base change along the map $\{x\} \lrar \sN\C$ (see explanations after Construction \ref{con:Un_CF}). Combined with Remark \ref{r:XKan}, we get a weak equivalence
\begin{equation}\label{eq:r:XKan}
\F(x)^{\op} \lrarsimeq \Un_{\C}^{\triangleleft}(\F) \times_{\sN\C} \{x\}
\end{equation}
whenever $\F(x)$ is a Kan complex.
\end{rem}

Let $x,y∈()$ be two objects of $$. Suppose we are given two vertices $μ∈(x)$ and $ν∈(y)$ regarded as vertices of $_^◃()$ (see Remark \ref{r:vertexedge}). We wish to give a convenient model for $^__^◃()(μ,ν)$ the space of right morphisms from $μ$ to $ν$ in $_^◃()$. Recall by definition that an $n$-simplex of $^__^◃()(μ,ν)$ is an $(n+1)$-simplex $T : Δ^n+1 _^◃()$ of $_^◃()$ such that $d_n+1T$ is degenerate on $μ$ and $TΔ^{n+1} = ν$. According to Construction \ref{con:Un_CF}, the data of $T$ consists of:

$∙$ An $(n+1)$-simplex $H : Δ^n+1 $ of $$ satisfying that $d_n+1H$ is degenerate on $x$ and $HΔ^{n+1} = y$. In other words, $H$ is nothing but an $n$-simplex of $^_(x, y)$. We also regard $H$ as a functor $ℭ[Δ^n+1] $.

$∙$ In addition, a natural transformation $t : 𝔇^◃_Δ^n+1 ∘H$ between simplicial functors $ℭ[Δ^n+1] _Δ$ satisfying that for every $i∈{0,⋯,n}$ the map $$t(i) : \mathfrak{D}^{\triangleleft}_{\Delta^{n+1}}(i) \lrar \F \circ H(i) = \F(x)$$ collapses its domain to the vertex $μ$ and that the initial vertex of $t(n + 1)$ agrees with $ν$. (Note that $t(n + 1)$ performs an $(n + 1)$-cube in $(y)$).

On other hand, we take $ρ_μ$ to be the composition $$ \rho_\mu : \Map_\C(x,y) \x{\F}{\lrar} \Map_{\Set_\Delta}(\F(x),\F(y)) \x{ev_\mu}{\lrar} \F(y)$$ with $ev_μ$ being the evaluation at $μ$. Now, by [[4], 2.2.2.13] there is a canonical isomorphism $^_(x, y) ≅_*^▹_(x,y)$. So we can form a canonical map
\begin{equation}\label{eq:mapH}
\Hom^{\rR}_{\sN\C}(x, y) \cong \Un_{*}^{\triangleright}\Map_\C(x,y) \x{\Un_{*}^{\triangleright}(\rho_\mu)}{\xrightarrow{\hspace*{1cm}}}  \Un_{*}^{\triangleright}\F(y) \; .
\end{equation}
Let us analyze this map. Let $H$ be an $n$-simplex of $^_(x, y)$ exhibited by a simplicial functor $H : ℭ[Δ^n+1] $. The data of $H$ consists of the maps of the form
$$H^{i} : \Map_{\mathfrak{C}[\Delta^{n+1}]}(i,n+1) \cong (\Delta^{1})^{\times (n-i)} \lrar \Map_\C(x,y) \; , \; i = 0,\cdots,n .$$
Observe that each map $H^i$ agrees with the restriction of $H^0$ to a certain $(n-i)$-cube. So the data of $H$ is fully displayed in that of $H^0$. It can be shown that $H^0$ performs a quasi $n$-simplex of $_(x,y)$ and hence, is an $n$-simplex of $_*^▹_(x,y)$ by Lemma \ref{l:quasisimplex}. Moreover, note that under the identification $^_(x, y) ≅_*^▹_(x,y)$, $H$ is simply identified to $H^0$. For each $i ∈{0,⋯,n}$, we denote by $H^i_*(μ) := H^i ∘ρ_μ$. Finally, we get that the map \eqref{eq:mapH} sends $H$ to $H^0_*(μ)$, (which performs a quasi $n$-simplex of $(y)$).

\begin{lem}\label{l:mappingUnF} There is a canonical isomorphism
$$ \Hom^{\rR}_{\Un_{\C}^{\triangleleft}(\F)}(\mu,\nu) \cong (\Un_{*}^{\triangleright}\F(y))_{\nu/} \times_{\Un_{*}^{\triangleright}\F(y)} \Hom^{\rR}_{\sN\C}(x, y) =: \mathbb{P}_{\mu,\nu} .$$
\end{lem}

\begin{proof} With having chosen $H \in \Hom^{\rR}_{\sN\C}(x, y)$, an $n$-simplex of $\mathbb{P}_{\mu,\nu}$ is a quasi $(n+1)$-simplex of $\F(y)$ which admits $\nu$ as its initial vertex and in addition, admits $H^{0}_*(\mu)$ as an end face. (To see this, we combine Lemma \ref{l:quasisimplex} with the above analyses).

On the other hand, with having chosen $H \in \Hom^{\rR}_{\sN\C}(x, y)$, an $n$-simplex of $\Hom^{\rR}_{\Un_{\C}^{\triangleleft}(\F)}(\mu,\nu)$ (as analyzed earlier) is a natural transformation $t : \mathfrak{D}^{\triangleleft}_{\Delta^{n+1}} \lrar \F \circ H$ satisfying that for every $i\in \{0,\cdots,n\}$ the map $t(i) : \mathfrak{D}^{\triangleleft}_{\Delta^{n+1}}(i) \lrar \F(x)$ collapses its domain to the vertex $\mu$ and that the initial vertex of $t(n + 1)$ agrees with $\nu$. In particular, the data of $t$ is fully enclosed in that of the map $$t(n+1) : \mathfrak{D}^{\triangleleft}_{\Delta^{n+1}}(n+1) = \sN\rP^{\triangleleft}(n+1)  \lrar \F(y)$$
subject to the following condition:

(**) for every $i\in \{0,\cdots,n\}$, the composition
$$ \Map_{\mathfrak{C}[\Delta^{n+1}]}(i,n+1) \times \mathfrak{D}^{\triangleleft}_{\Delta^{n+1}}(i) \x{\mathfrak{D}^{\triangleleft}_{\Delta^{n+1}}}{\xrightarrow{\hspace*{1cm}}} \mathfrak{D}^{\triangleleft}_{\Delta^{n+1}}(n+1) \x{t(n+1)}{\xrightarrow{\hspace*{1cm}}} \F(y) $$
agrees with the composition $\Map_{\mathfrak{C}[\Delta^{n+1}]}(i,n+1) \times \mathfrak{D}^{\triangleleft}_{\Delta^{n+1}}(i) \lrar \Map_{\mathfrak{C}[\Delta^{n+1}]}(i,n+1) \x{H^{i}_*(\mu)}{\xrightarrow{\hspace*{1cm}}} \F(y)$.

Note that the first composition corresponds to the restriction of $t(n+1)$ to $\sN \, \{A \in \rP^{\triangleleft}(n+1) \, | \, A \ni i \}$, which is an end face of the $(n+1)$-cube $\mathfrak{D}^{\triangleleft}_{\Delta^{n+1}}(n+1) = \sN\rP^{\triangleleft}(n+1)$.

When $i=0$, the condition (**) means that $H^{0}_*(\mu)$ agrees with the end face of $t(n+1)$ given by the restriction of itself to  $\sN \, \{A \in \rP^{\triangleleft}(n+1) \, | \, A \ni 0 \}$. In particular, for each $i\in \{0,\cdots,n\}$, $H^{i}_*(\mu)$ agrees with the restriction of $t(n+1)$ to $\sN \, \{A \in \rP^{\triangleleft}(n+1) \, | \, A \supseteq [0,i] \}$. So for every $i\in \{0,\cdots,n\}$, the condition (**) means that the end face of $t(n+1)$ given by the restriction $\restr{t(n+1)}{\sN \, \{A \in \rP^{\triangleleft}(n+1) \, | \, A \ni i \}}$ is degenerate on $\restr{t(n+1)}{\sN \, \{A \in \rP^{\triangleleft}(n+1) \, | \, A \supseteq [0,i] \}}$. This is equivalent to saying that $t(n+1)$ performs a quasi $(n+1)$-simplex of $\F(y)$. Therefore, by the first paragraph, the data of an $n$-simplex of $\Hom^{\rR}_{\Un_{\C}^{\triangleleft}(\F)}(\mu,\nu)$ is equivalent to that of an $n$-simplex of $\mathbb{P}_{\mu,\nu}$. The obtained identification is indeed compatible with simplicial structures.
\end{proof}

\begin{rem} There is a canonical embedding $\Un_{*}^{\triangleright}(\F(y)_{\nu/}) \lrar (\Un_{*}^{\triangleright}\F(y))_{\nu/}$ established as follows. By Lemma \ref{l:quasisimplex}, an $n$-simplex of $\Un_{*}^{\triangleright}(\F(y)_{\nu/})$ is a quasi $n$-simplex of $\F(y)_{\nu/}$, which can be identified with a map $$\gamma : \Delta^{0} \star (\Delta^{1})^{\times\{1,\cdots,n\}} \lrar \F(y)$$ (where $``\star$'' stands for the \textbf{join operation}) satisfying that for every $i \in \{1,\cdots,n\}$ the restriction $\restr{\gamma}{\Delta^{0} \star T_i}$ is degenerate on $\restr{\gamma}{\Delta^{0} \star T^{\circ}_i}$ and that $\restr{\gamma}{\Delta^{0}}$ agrees with $\nu$. We then associates to $\gamma$ the composite map
$$ \widetilde{\gamma} : (\Delta^{1})^{\times\{1,\cdots,n+1\}} \lrar \Delta^{0} \star (\Delta^{1})^{\times\{1,\cdots,n\}} \x{\gamma}{\lrar} \F(y) $$
in which the first map refers to the canonical way of collapsing the $(n+1)$-cube to its subset given by joining the initial vertex with an end face. The map $\widetilde{\gamma}$ is indeed a quasi $(n+1)$-simplex of $\F(y)$ with $\nu$ as its initial vertex, and hence is an $n$-simplex of $(\Un_{*}^{\triangleright}\F(y))_{\nu/}$.
\end{rem}

By the above remark and Remark \ref{r:XKan}, we can form a composite map
\begin{equation}\label{eq:modelmappingspace}
\F(y)_{\nu/} \times_{\F(y)} \Map_\C(x,y)  \lrar  \Un_{*}^{\triangleright}(\F(y)_{\nu/}) \times_{\Un_{*}^{\triangleright}\F(y)} \Un_{*}^{\triangleright}\Map_\C(x,y)  \lrar \mathbb{P}_{\mu,\nu}
\end{equation}

Suppose now that $$ is fibrant and $: _Δ$ is levelwise fibrant. Then the map $_^◃() $ is a left fibration and $$ is an $∞$-category. So $_^◃()$ is an $∞$-category as well. Our main interest in this subsection is the following:
\begin{prop}\label{p:mappingunCF}  Let $x,y\in \Ob(\C)$ be two objects of $\C$. Given two vertices $\mu \in \F(x)$ and $\nu \in \F(y)$ regarded as vertices of $\Un_{\C}^{\triangleleft}(\F)$, there is a homotopy equivalence
$$ \{\nu\} \times_{\F(y)}^{\h} \Map_\C(x,y) \lrarsimeq \Map_{\Un_{\C}^{\triangleleft}(\F)}(\mu,\nu) .$$

\end{prop}
\begin{proof} We make use of the pullback $\F(y)_{\nu/} \times_{\F(y)} \Map_\C(x,y)$ as a model for the left hand side. Since $\F(y)$ and $\Map_\C(x,y)$ are Kan complexes, that pullback is Kan as well. It implies that the first map of the composition \eqref{eq:modelmappingspace} is a weak equivalence (see Remark \ref{r:XKan}). The second map of \eqref{eq:modelmappingspace} is also a weak equivalence, since $\Un_{*}^{\triangleright}(\F(y)_{\nu/})$ and $(\Un_{*}^{\triangleright}\F(y))_{\nu/}$ are both contractible. So the map \eqref{eq:modelmappingspace} is a weak equivalence. We deduce then by using Lemma \ref{l:mappingUnF}.
\end{proof}

\begin{rem} An analogue holds in the contravariant case. Let $\F : \C^{\op} \lrar \Set_\Delta$ be a simplicial presheaf on $\C$. Let $x,y\in \Ob(\C)$ be two objects of $\C$, and suppose we are given two vertices $\mu \in \F(x)$ and $\nu \in \F(y)$ regarded as vertices of $\Un_{\C}^{\triangleright}(\F)$. We take $\rho_\nu$ to be the composite map
$$ \rho_\nu : \Map_\C(x,y) \x{\F}{\lrar} \Map_{\Set_\Delta}(\F(y),\F(x)) \x{ev_\nu}{\lrar} \F(x)$$ with $ev_\nu$ being the evaluation at $\nu$.
When $\C$ is fibrant and $\F$ is levelwise fibrant (so that $\Un_{\C}^{\triangleright}(\F)$ is an $\infty$-category), there is a homotopy equivalence
$$ (\F(x)_{\mu/} \times_{\F(y)} \Map_\C(x,y))^{\op} \lrarsimeq \Map_{\Un_{\C}^{\triangleright}(\F)}(\mu,\nu) .$$
\end{rem}

\smallskip

\subsection{Twisted arrow $\infty$-categories of simplicial operads}

\smallskip

Let $ℰ$ be an ordinary category. The \textit{(covariant) twisted arrow category} $(ℰ)$ is by definition the category whose objects are the morphisms of $ℰ$ and such that maps from $f:xy$ to $f':x'y'$ are given by commutative diagrams of the form
\begin{equation}\label{eq:twisted}
\begin{tikzcd}[row sep=2em, column sep =2em]
x \arrow[d, "f"'] & x' \arrow[l] \arrow[d, "f'"] & \\
y \arrow[r] & y'
\end{tikzcd}
\end{equation}
This classical notion was generalized into the $∞$-categorical framework due to Lurie [[3], \S 5.2.1]. For a given $∞$-category $$, the \textbf{twisted arrow category} $()$ is the $∞$-category whose $n$-simplices are the $(2n+1)$-simplices of $$.	In particular, objects of $()$ are the morphisms of $$. A map from $f:x y$ to $f':x'y'$ can be also depicted as a diagram of the type \eqref{eq:twisted} commutative up to a chosen homotopy.

Furthermore, for a given fibrant simplicial category $$, the twisted arrow $∞$-category of $$ is simply $() := ()$ the twisted arrow $∞$-category of the nerve of $$. It turns out that $()$ can be represented as the unstraightening of a certain simplicial copresheaf. This phenomenon motivates us to establish \textit{twisted arrow $\infty$-categories of (fibrant) simplicial operads}. Let us see how it arises.

\smallskip

We will need the following notations and conventions:

- Let $S$ be a simplicial set. We denote by $(_Δ^)_/S$ the \textbf{covariant model category of simplicial sets over} $S$ whose cofibrations are monomorphisms and whose fibrant objects are left fibrations over $S$. (See [[4], \S 2.1.4] for more details).

- Let $$ be a simplicial category. Recall that the category of $$-bimodules can be represented as $(^×, _Δ)$ the category of simplicial functors  $^×_Δ$. Moreover, under this identification, the functor $_: ^×_Δ$, $(x,y) ↦_(x,y)$ is nothing but $$, viewed as a bimodule over itself. We endow $(^×, _Δ)$ with the projective model structure.

- For the remainder, we consider only the covariant unstraightening construction. So we will simply write $_(-)$ standing for the covariant unstraightening of simplicial functors $_Δ$.

\smallskip

Suppose now that $$ is a fibrant simplicial category. According to [[3], Proposition 5.2.1.11], the unstraightening functor
\begin{equation}\label{eq:uncat}
\Un_{\C^{\op}\times\C} : \Fun(\C^{\op}\times\C , \Set_\Delta) \lrarsimeq (\Set_\Delta^{\cov})_{/\sN\C^{\op}\times\sN\C},
\end{equation}
which is a right Quillen equivalence, identifies $_$ to $()$. In other words, the unstraightening of $$ (viewed as a bimodule over itself) is exactly a model for $()$.

We now fix $$ to be a fibrant simplicial $C$-colored operad. Recall from Section \ref{s:infbimod} that there is a canonical isomorphism $() ≅(Ib^𝒫,_Δ)$ between the categories of infinitesimal $$-bimodules and simplicial functors $Ib^𝒫 _Δ$. The unstraightening construction gives us a right Quillen equivalence
\begin{equation}\label{eq:unop}
\Un_{\textbf{Ib}^{\mathcal{P}}}: \IbMod(\P) = \Fun(\textbf{Ib}^{\mathcal{P}},\Set_\Delta) \lrarsimeq (\Set_\Delta^{\cov})_{/\sN(\textbf{Ib}^{\mathcal{P}})}.
\end{equation}
Since $$ is fibrant, it implies that $Ib^𝒫$ is a fibrant simplicial category, and hence $(Ib^𝒫)$ is an $∞$-category. Moreover, the functor
$$\P : \textbf{Ib}^{\mathcal{P}} \lrar \Set_\Delta \; , \; (c_1,\cdots,c_m;c) \mapsto \P(c_1,\cdots,c_m;c),$$ which encodes the data of $$ as an infinitesimal bimodule over itself, is levelwise fibrant. So we get a left fibration $_Ib^𝒫() (Ib^𝒫)$, and hence $_Ib^𝒫()$ is in particular an $∞$-category.

Along the passage from simplicial categories to simplicial operads, the notion of categorical bimodules should correspond to that of operadic infinitesimal bimodules, which can be thought of as the linearization of the usual notion of operadic bimodules. With having this logic in mind, we propose the following concept.

\begin{dfn} The \textbf{twisted arrow $\infty$-category of} $\P$, denoted by $\Tw(\P)$, is defined to be $\Tw(\P) := \Un_{\textbf{Ib}^{\mathcal{P}}}(\P)$ the unstraightening of the simplicial functor $\P : \textbf{Ib}^{\mathcal{P}} \lrar \Set_\Delta$ or alternatively, unstraightening of $\P$ regarded as an infinitesimal bimodule over itself.
\end{dfn}

\begin{examples}\label{ex:twistcom&ass} When $\P$ is discrete then $\Tw(\P)$ is isomorphic to the nerve of an ordinary category. In this situation, we will identify $\Tw(\P)$ to the corresponding ordinary category and refer to it as the \textbf{twisted arrow category of $\P$}.  For example, it is not hard to show that the twisted arrow category of the \textbf{commutative operad} is equivalent to $\Fin_*^{\op}$. We will see that the twisted arrow category of the \textbf{associative operad} is equivalent to the simplex category $\Delta$ (cf. Proposition \ref{p:twistass}).
\end{examples}

\begin{prop}\label{p:Twisgood} The construction $\Tw(-)$  determines a homotopy invariant from fibrant simplicial operads to $\infty$-categories.
\end{prop}
\begin{proof} Let $f:\P\rar\Q$ be a map between fibrant simplicial operads. By the compatibility of the unstraightening functor with taking base change along $f:\P\rar\Q$, we obtain the induced map $\Tw(f):\Tw(\P) \lrar\Tw(\Q)$ fitting into the following Cartesian square of $\infty$-categories
$$ \xymatrix{
\Tw(\P)  \ar[r]\ar[d] & \Tw(\Q) \ar[d] \\
\sN(\textbf{Ib}^{\mathcal{P}}) \ar[r] & \sN(\textbf{Ib}^{\mathcal{Q}}) \\
Note that this square is already homotopy Cartesian (with respect to the Joyal model structure), due to the fact that the right vertical map is a left fibration.

We are showing that the map $\Tw(f):\Tw(\P) \lrar\Tw(\Q)$ is an equivalence when provided that $f:\P\rar\Q$ is a weak equivalence. We first show that the induced map $\textbf{Ib}^{f} : \textbf{Ib}^{\mathcal{P}}\lrar \textbf{Ib}^{\mathcal{Q}}$ is a weak equivalence of simplicial categories (i.e., a Dwyer-Kan equivalence). It is clear  by construction that $\textbf{Ib}^{f}$ is a levelwise weak equivalence. Hence it remains to show that $\textbf{Ib}^{f}$ is essentially surjective. Suppose given an object $(d_1,\cdots,d_n;d_0)$ of $\textbf{Ib}^{\mathcal{Q}}$. Since the underlying simplicial functor  $f_1 : \P_1\lrar\Q_1$ of $f$ is essentially surjective, for each $i\in \{0,\cdots,n\}$ there exists an object $c_i$ of $\P$ together with an isomorphism $\theta_i : f(c_i) \lrarsimeq d_i$ in the homotopy category of $\Q_1$. The morphisms $\theta_i$'s together form a morphism $ \theta : ( f(c_1),\cdots,f(c_n);f(c_0) ) \lrar (d_1,\cdots,d_n;d_0)$ in $\textbf{Ib}^{\mathcal{Q}}$. It can then be verified by definition that $\theta$ is an isomorphism in the homotopy category of $\textbf{Ib}^{\mathcal{Q}}$. We just proved that $\textbf{Ib}^{f}$ is a weak equivalence (between fibrant simplicial categories). So the map $\sN(\textbf{Ib}^{\mathcal{P}}) \lrar \sN(\textbf{Ib}^{\mathcal{Q}})$ is an equivalence of $\infty$-categories. This fact, together with the first paragraph, proves that the map $\Tw(f):\Tw(\P)\lrar\Tw(\Q)$ is indeed an equivalence
\end{proof}

Again, let $$ be a fibrant simplicial $C$-colored operad. Due to the previous subsection, we can describe the simplicial structure of $()$ as follows.

\begin{cons}\label{cons:TwP} The data of an $n$-simplex of $\Tw(\P)$ consists of

$\bullet$ an $n$-simplex  $\varphi\in\sN(\textbf{Ib}^{\mathcal{P}})$, i.e., a functor $\varphi : \mathfrak{C}[\Delta^{n}]\lrar \textbf{Ib}^{\mathcal{P}}$, and

$\bullet$ a natural transformation $t:\mathfrak{D}^{\triangleleft}_{\Delta^{n}}\lrar \P \circ \varphi$ between simplicial functors $\mathfrak{C}[\Delta^{n}] \lrar \Set_\Delta$.

\noindent For each map $\delta : \Delta^{m}\lrar\Delta^{n}$, the simplicial structure map $\Tw(\P)_n\lrar\Tw(\P)_m$ is induced by the natural transformation  $\mathfrak{D}^{\triangleleft}_{\delta} : \mathfrak{D}^{\triangleleft}_{\Delta^{m}} \lrar \mathfrak{D}^{\triangleleft}_{\Delta^{n}} \circ \mathfrak{C}[\delta]$, as in Construction \ref{con:Un_CF}.
\end{cons}

To analyze the structure of $()$, we will need the following notations.

\begin{notn} We always denote the permutations by listing their values. For instance, $\sigma=[i_1,\cdots,i_n]$ refers to the permutation $\sigma\in \Sigma_n$ with $\sigma(k)=i_k$.
\end{notn}

\begin{notns}\label{no:pointedmap} Let $f : \l n \r \lrar \l m \r$ be a map in $\Fin_*$. For each $s\in\{0,1,\cdots,m\}$, we let $[f^{-1}(s)]$ denote the increasing sequence of the elements of $f^{-1}(s)$,  written as $ [f^{-1}(s)]  = [ i_1^{s}<\cdots<i_{k_s}^{s} ]$. (Of course, this could be empty). Then $f$ can be represented by the sequence obtained by concatenating those sequences for $s = 1,\cdots,m$, written as:
$$ [f^{-1}(1) \; | \; f^{-1}(2) \; | \; \cdots \; | \; f^{-1}(m)]  = [ i_1^{1} < \cdots < i_{k_1}^{1} \; | \; i_1^{2} < \cdots < i_{k_2}^{2} \; | \; \cdots \; | \; i_1^{m} < \cdots < i_{k_m}^{m} ]  ,$$
or alternatively, by the extended sequence $[f^{-1}(1) \; | \; f^{-1}(2) \; | \; \cdots \; | \; f^{-1}(m) \; | \; f^{-1}(0)^{\circ}]$ formed in the same manner, in which $ [f^{-1}(0)^{\circ}] := [f^{-1}(0)] - \{0\}$. Moreover, we denote by $\sigma_f$ the permutation $[f^{-1}(1) \; | \; f^{-1}(2) \; | \; \cdots \; | \; f^{-1}(m) \; | \; f^{-1}(0)^{\circ}] \in \Sigma_n$.
\end{notns}

\begin{notn}\label{no:fortwosimplex} Let $\ovl{c}:=(c_1,\cdots,c_m;c)$ and $\ovl{d}:=(d_1,\cdots,d_n;d)$ be two $C$-sequences and $f:\langle n \rangle \lrar \langle m \rangle$ a map in $\Fin_*$. Given a vertex $$\alpha = (\alpha_0, \alpha_1,\cdots,\alpha_m) \in \Map^{f}_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{d})$$ with $\alpha_0\in \mathcal{P}\left (c,\{d_j\}_{j\in f^{-1}(0)};d \right )$ and $\alpha_i\in\mathcal{P} \left (\{d_j\}_{j\in f^{-1}(i)};c_i \right )$ ($i=1,\cdots,m$) (see Construction \ref{catencodinginfbimod}), we denote by  $\alpha^{*} :  \mathcal{P}(\ovl{c}) \lrar \mathcal{P}(\ovl{d})$ the image of $\alpha$ under the map $\Map_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{d}) \lrar \Map_{\Set_\Delta}(\mathcal{P}(\ovl{c}) , \mathcal{P}(\ovl{d}))$ which is part of the simplicial functor structure of $\P : \textbf{Ib}^{\P} \lrar \Set_\Delta$. By construction, for each simplex $\theta\in \mathcal{P}(\ovl{c})$, we have that
$$\alpha^{*}(\theta)=(\,\alpha_0 \circ_1 \theta \circ (\alpha_1,\cdots,\alpha_m)\,)^{\sigma_f^{-1}} \in\mathcal{P}(\ovl{d}) $$
the action of $\sigma_f^{-1} \in \Sigma_n$ on $\alpha_0 \circ_1 \theta \circ (\alpha_1,\cdots,\alpha_m)$, where $``$($\circ_1$) $\circ$'' refers to the (partial) composition.
\end{notn}

\smallskip

Unwinding definition we will see that $\Tw(\P)$ indeed looks like something obtained by twisting $``$multiarrows'' of $\P$.

$\bullet$ Objects of $\Tw(\P)$ are precisely the \textbf{operations} of $\P$ (i.e., the vertices of the spaces of operations of $\P$).

$\bullet$ Let $\mu \in \P(\ovl{c})$ and $\nu\in\P(\ovl{d})$ be two operations of $\P$, the data of a morphism (edge) $\mu\rar \nu$ in $\Tw(\P)$ consists of

- a map $f:\langle n \rangle \lrar \langle m \rangle$ in $\Fin_*$,

- a tuple of operations  $\alpha = (\alpha_0, \alpha_1,\cdots,\alpha_m) \in \Map^{f}_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{d})$, and

- an edge $t : \nu \lrar \alpha^{*}(\mu)$ in $\P(\ovl{d})$, viewed as a homotopy from $\nu$ to $\alpha^{*}(\mu)$.

\noindent By convention, we will write $(f,\alpha,t) : \mu \lrar \nu $ standing for such a typical morphism. It is convenient to depict this morphism as the diagram of the form
\begin{equation}\label{eq:1simplex}
\begin{tikzcd}[row sep=3.5em, column sep =3em]
(c_1, \cdots ,c_m) \arrow[d, "\mu"'] & (d_1, \cdots ,d_n) \arrow[l, "(\alpha_1 \, \cdots \, \alpha_m )"'] \arrow[d, "\nu"] & \\
(c) \arrow[r, "\alpha_0"'] & (d)
\end{tikzcd}
\end{equation}
which is $``$commutative up to a chosen homotopy''.

$\bullet$ In general, a $k$-simplex of $\Tw(\P)$ is the one, formally, depicted as the composition of $k$ squares of the type \eqref{eq:1simplex} equipped with a collection of homotopies, homotopies between homotopies, and so forth.

\smallskip

We saw above the simplicial structure of $\Tw(\P)$. So it remains to understand the $\infty$-categorical structure of $\Tw(\P)$. As usual, this is best understood by reformulating equivalences in $\Tw(\P)$ and describing its mapping spaces. We indeed hold these in hand, expressed by the two propositions below.

Again, let $\mu \in \P(\ovl{c})$ and $\nu\in\P(\ovl{d})$ be two operations of $\P$, regarded as objects of $\Tw(\P)$.
\begin{prop}\phantomsection\label{p:eqvlTwP} A morphism $(f,\alpha,t) : \mu \lrar \nu$ is an equivalence in $\Tw(\P)$ if and only if the following conditions hold:
\begin{enumerate}
\item $f : \l n \r \lrar \l m \r$ is bijective.
\item There exist $\beta\in\Map^{f^{-1}}_{\textbf{Ib}^{\P}}(\ovl{d} , \ovl{c})$, an edge $t' : \mu \lrar \beta^{*}(\nu)$ and together with two edges $$ [h: id_{\ovl{c}} \lrar \beta\alpha] \in \Map^{\Id_{\l m \r}}_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{c}) \; \; , \; \; [h': id_{\ovl{d}} \lrar \alpha\beta] \in \Map^{\Id_{\l n \r}}_{\textbf{Ib}^{\P}}(\ovl{d} , \ovl{d}) $$ which are all subject to the existence of two 2-simplices of the forms
\begin{equation}\label{eq:twotriangles}
\begin{tikzcd}[column sep=1.5em]
& \beta^{*}(\nu) \arrow{dr}{\beta^{*}(t)} \\
\mu \arrow{ur}{t'} \arrow{rr}{h^{*}|_{\{\mu\}\times\Delta^{1}}} && \beta^{*}\alpha^{*}(\mu)
\end{tikzcd}\qquad%
\begin{tikzcd}[column sep=1.5em]
& \alpha^{*}(\mu) \arrow{dr}{\alpha^{*}(t')} \\
\nu \arrow{ur}{t} \arrow{rr}{(h')^{*}|_{\{\nu\}\times\Delta^{1}}} && \alpha^{*}\beta^{*}(\nu)
\end{tikzcd}
\end{equation}
belonging to  $\P(\ovl{c})$ and $\P(\ovl{d})$, respectively. In this item, $h^{*}$ (and similarly, $(h')^{*}$) is given by the composition
$$\P(\ovl{c}) \times \Delta^{1} \x{\Id_{\P(\ovl{c})}\times h}{\xrightarrow{\hspace*{1.5cm}}} \P(\ovl{c}) \times \Map^{\Id_{\l m \r}}_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{c}) \x{\P}{\lrar} \P(\ovl{c}).$$
\end{enumerate}
\end{prop}
\begin{proof} The proof is straightforward, once one knows that the data of a 2-simplex of $\Tw(\P)$ of the form
\begin{center}
\begin{tikzcd}[column sep=1.5em]
& \nu \arrow{dr}{(g,\beta,t')} \\
\mu \arrow{ur}{(f,\alpha,t)} \arrow{rr}{s_0\mu} && \mu \; ,
\end{tikzcd}
\end{center}
with having (necessarily) required $fg = \Id_{\l m \r}$, is equivalent to the choice of an edge $h: id_{\ovl{c}} \lrar \beta\alpha$ in $\Map^{\Id_{\l m \r}}_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{c})$ and in addition, a square in $\P(\ovl{c})$ of the form
$$ \xymatrix{
\mu  \ar^{s_0\mu}[r]\ar_{t'}[d] & \mu \ar^{h^{*}|_{\{\mu\}\times\Delta^{1}}}[d] \\
\beta^{*}(\nu) \ar_{\beta^{*}(t)}[r] & \alpha^{*}\beta^{*}(\nu) \; . \\
Moreover, since $\P(\ovl{c})$ is Kan, this square is equivalent to the choice of a 2-simplex of the form \eqref{eq:twotriangles} (the first one).
\end{proof}

We take $\rho_\mu$ to be the composite map
$$ \rho_\mu : \Map_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{d}) \x{\P}{\lrar} \Map_{\Set_\Delta}(\mathcal{P}(\ovl{c}) , \mathcal{P}(\ovl{d})) \x{ev_\mu}{\lrar} \mathcal{P}(\ovl{d}) $$
with $ev_\mu$ being the evaluation at $\mu$.

\begin{prop}\label{p:mappingtwp} There is a canonical homotopy equivalence
$$ \{\nu\} \times^{\h}_{\P(\ovl{d})}\Map_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{d}) \lrarsimeq \Map_{\Tw(\P)}(\mu , \nu) .$$
\end{prop}
\begin{proof} This follows immediately by Proposition \ref{p:mappingunCF}.
\end{proof}

\begin{rem}\label{r:mappingcomponent} Note that there is a canonical map of $\infty$-categories $\Tw(\P) \lrar \sN(\Fin_*^{\op})$ factoring through $\sN(\textbf{Ib}^{\mathcal{P}})$. Consider the induced map $\Map_{\Tw(\P)}(\mu , \nu) \lrar \Hom_{\Fin_*^{\op}}(\langle m \rangle , \langle n \rangle)$. For each map $f : \langle n \rangle \lrar \langle m \rangle$ in $\Fin_*$, we denote by $\Map^{f}_{\Tw(\P)}(\mu , \nu)$ the component of $\Map_{\Tw(\P)}(\mu , \nu)$ over $f$, so that we can write
$$ \Map_{\Tw(\P)}(\mu , \nu) = \bigsqcup_{\langle n \rangle \x{f}{\lrar} \langle m \rangle} \Map^{f}_{\Tw(\P)}(\mu , \nu) .$$
By the above proposition, there is a homotopy equivalence
$$ \{\nu\} \times^{\h}_{\P(\ovl{d})}\Map^{f}_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{d}) \lrarsimeq \Map^{f}_{\Tw(\P)}(\mu , \nu)  .$$
\end{rem}

In what follows, we will survey some local properties of $\Tw(\P)$ using the two propositions above.

\begin{cor} \label{e:skeletontw} Let $\mu\in \P(c_1,\cdots,c_m;c)$ and $\sigma\in\Sigma_m$ be given. Then there is a canonical equivalence $\mu \lrarsimeq \mu^{\sigma}$ in $\Tw(\P)$.
\end{cor}
\begin{proof} We take a canonical edge $(f,\alpha,t) : \mu\lrar\mu^{\sigma}$ as follows. The map $f : \l m \r  \lrar \l m \r$ agrees with $\sigma$ on $\{1,\cdots,m\}$. Then, we take $\alpha$ to be the tuple of unit operations $$\alpha := (\id_c,\id_{c_1}\cdots,\id_{c_m}) \in \Map^{\sigma}_{\textbf{Ib}^{\P}}( (c_1,\cdots,c_m;c) , (c_{\sigma(1)},\cdots,c_{\sigma{(m)}};c) ).$$ By construction, the induced map $$\alpha^{*} : \P(c_1,\cdots,c_m;c) \lrar \P(c_{\sigma(1)},\cdots,c_{\sigma(m)};c)$$ is nothing but the map defined by the action of the permutation $\sigma$ on $\P$ (see the proof of Proposition \ref{rewriteinfinitesimal1}). In particular, we get that $\alpha^{*}(\mu)=\mu^{\sigma}$. Finally, we take $t := s_0\mu^{\sigma}$. Using Proposition \ref{p:eqvlTwP}, we can readily verify that the obtained edge $(f,\alpha,t)$ is indeed an equivalence in $\Tw(\P)$.
\end{proof}

By convention, a simplicial operad is said to be $k$-\textbf{connected} if all its spaces of operations are $k$-connected. For example, the \textbf{little $n$-cubes operad} $\E_n$ is $(n-2)$-connected (cf., e.g., [[11], Corollary 5.1.11]).  As before, let $\mu \in \P(\ovl{c})$ and $\nu\in\P(\ovl{d})$ be two operations of $\P$, regarded as objects of $\Tw(\P)$.

\begin{cor} Suppose we are given an integer $k$ and suppose further that $\P$ is $k$-connected.  Then the induced map
\begin{equation}\label{eq:kconnected}
\Map_{\Tw(\P)}(\mu , \nu) \lrar \Hom_{\Fin_*^{\op}}(\langle m \rangle,\langle n \rangle)
\end{equation}
is $k$-connected. In particular, whenever $k\geqslant1$, the map $\Tw(\P) \lrar \sN(\Fin_*^{\op})$ induces an equivalence $\Ho(\Tw(\P)) \lrarsimeq \Fin_*^{\op}$ between (ordinary) homotopy categories.
\end{cor}
\begin{proof}  For each map $f : \langle n \rangle \lrar \langle m \rangle$ in $\Fin_*$, by Remark \ref{r:mappingcomponent} we have a fiber sequence
$$ \Map^{f}_{\Tw(\P)}(\mu , \nu) \lrar \Map^{f}_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{d}) \lrar \P(\ovl{d}) .$$
By assumption, both the second and third terms are $k$-connected. It implies that $\Map^{f}_{\Tw(\P)}(\mu , \nu)$ is $(k-1)$-connected, and hence the map \eqref{eq:kconnected} is indeed $k$-connected.

Consequently, when $k\geqslant1$ we have that $$ \pi_0\Map_{\Tw(\P)}(\mu , \nu) \cong \Hom_{\Fin_*^{\op}}(\langle m \rangle,\langle n \rangle) .$$
In particular, this implies that any two operations of $\P$ that are of the same arity are equivalent as objects of $\Tw(\P)$. The proof is therefore completed.
\end{proof}

We find a somewhat large class of simplicial operads whose twisted arrow $\infty$-categories admit terminal objects.
\begin{dfn}\label{d:unihoconnected} A simplicial operad is said to be \textbf{homotopy unital} if all its spaces of unary operations are weakly contractible. Furthermore, a simplicial operad is said to be \textbf{unitally homotopy connected} if it is homotopy unital with weakly contractible spaces of 1-ary operations.
\end{dfn}

Typical examples for this definition include the little cubes operads $\E_n$ for $n=0,\cdots,\infty$.

\begin{cor}\label{l:terminal} Let $\P$ be a fibrant and homotopy unital simplicial operad. Let $d$ be a color of $\P$ and $\mu_d \in \P(d)$ an unary operation of $\P$. Then $\mu_d$ is a terminal object of $\Tw(\P)$ if and only if for every color $c$, the space $\P(c;d)$ is contractible. Consequently, if $\P$ is fibrant and unitally homotopy connected then $\Tw(\P)$ admits terminal objects being precisely the unary operations of $\P$.
\end{cor}
\begin{proof} By definition, $\mu_d$ is a terminal object of $\Tw(\P)$ precisely if for every operation $\mu\in \P$ the mapping space $\Map_{\Tw(\P)}(\mu,\mu_d)$ is contractible. Given any operation $\mu\in \P(c_1,\cdots,c_m ; c)$, it suffices to show that there is a homotopy equivalence of spaces:
$$ \P(c;d) \simeq \Map_{\Tw(\P)}(\mu,\mu_d) .$$
By Proposition \ref{p:mappingtwp} there is a  homotopy equivalence
$$ \{\mu_d\}\times^{h}_{\P(d)}\Map_{\textbf{Ib}^{\P}}( (c_1,\cdots,c_m ; c) , (d) ) \lrarsimeq  \Map_{\Tw(\P)}(\mu,\mu_d) .$$
Since $\P(d)$ is contractible by assumption, the homotopy pullback is equivalent to $\Map_{\textbf{Ib}^{\P}}( (c_1,\cdots,c_m ; c) , (d) )$. Furthermore, note that $$ \Map_{\textbf{Ib}^{\P}}( (c_1,\cdots,c_m ; c) , (d) ) = \P(c;d) \times \P(c_1) \times \cdots \times \P(c_m) ,$$ which is therefore weakly equivalent to $ \P(c;d)$, by assumption again. So we get indeed a homotopy equivalence $\P(c;d) \simeq \Map_{\Tw(\P)}(\mu,\mu_d)$.
\end{proof}

Finally, for more illustration, we prove the following:
\begin{prop}\label{p:twistass} There is a canonical equivalence $\varphi : \Delta \lrarsimeq \Tw(\A ss)$ between the simplex category and the twisted arrow category of the associative operad $\A ss$.
\end{prop}

To this end, we first revisit some basic constructions.
\begin{cons}\label{cons:functoriota} There is a canonical functor $\iota : \Delta \lrar \Fin_*^{\op}$ (which is essentially used to define underlying cosimplicial spaces of \textbf{gamma spaces}) defined by sending $[n]\in \Delta$ to $\l n \r\in\Fin_*^{\op}$ and by giving the natural maps $\iota_{m,n} : \Hom_{\Delta}([m] , [n]) \lrar \Hom_{\Fin_*}(\l n \r , \l m \r)$ as follows. Given a map $f : [m] \lrar [n]$, let us identify $f$ with the increasing sequence $[j_1<\cdots<j_k]$ of its values together with a tuple $(p_1,\cdots,p_k)$ with $p_r$ being the cardinality of the fiber $f^{-1}(j_r)$. The map $\iota_{m,n}(f) : \l n \r \lrar \l m \r$ is given by listing its nonempty fibers as follows:
$$ \iota_{m,n}(f)^{-1}(p_1) = \{j_1+1,\cdots,j_2\} \, \; , \, \; \iota_{m,n}(f)^{-1}(p_1+p_2) = \{j_2+1,\cdots,j_3\} \, \; , \cdots ,$$
$$ \iota_{m,n}(f)^{-1}(p_1+p_2+\cdots+p_{k-1}) =\{j_{k-1}+1,\cdots,j_k\} \, \; , \, \; \iota_{m,n}(f)^{-1}(0) = \{0,1,\cdots,n\} - \{j_1+1,\cdots,j_k\} .$$
\end{cons}

\smallskip

\begin{rem}\label{r:twass} It is straightforward to verify the following observations:

(1) The image $\img(\iota_{m,n})\subseteq \Hom_{\Fin_*}(\l n \r , \l m \r)$ of $\iota_{m,n}$ consists of precisely those maps $g$ such that $[g^{-1}(1) \; | \; g^{-1}(2) \; | \; \cdots \; | \; g^{-1}(m)]$ (cf. Notations \ref{no:pointedmap}) is either empty or forms a sequence of consecutive natural numbers.

(2) Let $\Hom^{\const}_{\Delta}([m] , [n])\subseteq \Hom_{\Delta}([m] , [n])$ denote the subset consisting of the constant maps and let $\Hom^\circ_{\Delta}([m] , [n])$ denote the complement of the previous one in $\Hom_{\Delta}([m] , [n])$. Likewise, we denote by $\const_0\in\Hom_{\Fin_*}(\l n \r , \l m \r)$ the unique constant map (with value $0\in \l m \r$) and let $\img^{\circ}(\iota_{m,n})$ be the complement $ \img(\iota_{m,n}) - \{\const_0\}$.  Observe then that the restriction of $\iota_{m,n}$ to $\Hom^\circ_{\Delta}([m] , [n])$ induces a natural bijection $ \iota^{\circ}_{m,n} : \Hom^\circ_{\Delta}([m] , [n]) \lrarsimeq  \img^{\circ}(\iota_{m,n})$.
\end{rem}

Recall that the associative operad $\A ss$ is the single-colored operad whose set of $n$-ary operations is given by $\A ss(n)=\Sigma_n$ for $n\geqslant0$, equipped with the canonical right action of $\Sigma_n$ on itself. The composition is defined by concatenating linear orders.

\begin{proof}[\underline{Proof of Proposition~\ref{p:twistass}}] By Corollary \ref{e:skeletontw}, $\Tw(\A ss)$ admits a skeleton whose objects are the trivial permutations $\mu_n:=[1,\cdots,n] \in \Sigma_n$, $n\geqslant 0$. We define $\varphi$ on objects by sending $[n]\in\Delta$ to $\mu_n$. It remains to establish natural isomorphisms of the form
$$ \varphi_{m,n} :  \Hom_{\Delta}([m] , [n]) \lrarsimeq  \Hom_{\Tw(\A ss)} (\mu_m , \mu_n) .$$
Let us analyze the right hand side. By definition, we write $$ \Hom_{\Tw(\A ss)} (\mu_m , \mu_n) = \underset{\l n \r \overset{f}{\rar} \l m \r}{\bigsqcup} \A_{f} $$
where $\A_{f}\subseteq\Hom^{f}_{\textbf{Ib}^{\A ss}}( (c_1,\cdots,c_m ; c) , (d_1,\cdots,d_n ; d) )$ is the subset consisting of those $\alpha = (\alpha_0 , \alpha_1 , \cdots , \alpha_m )$ such that $\alpha^{*}(\mu_m)=\mu_n$ (see Notation \ref{no:fortwosimplex}). Unwinding definition, the latter is equivalent to the equation
\begin{equation}\label{eq:alpha}
\alpha_0 \circ_1 \mu_m \circ (\alpha_1 , \cdots , \alpha_m) = [f^{-1}(1) \; | \; f^{-1}(2) \; | \; \cdots \; | \; f^{-1}(m) \; | \; f^{-1}(0)^{\circ}]
\end{equation}
Observe that there is a unique choice of $(\alpha_1 , \cdots , \alpha_m)$ such that this equation possibly admits solutions, being precisely $(\mu_{h(1)} , \cdots , \mu_{h(m)})$, i.e., the tuple of trivial permutations with $h(i)$ referring to the arity of $\alpha_i$. Thus, by comparing the two sides of \eqref{eq:alpha} after substituting $(\alpha_1 , \cdots , \alpha_m) = (\mu_{h(1)} , \cdots , \mu_{h(m)})$ back to it, we realize that the set $\A_{f}$ is nonempty only if the sequence $[f^{-1}(1) \; | \; f^{-1}(2) \; | \; \cdots \; | \; f^{-1}(m)]$, whenever it is nonempty, forms a sequence of consecutive natural numbers. But this condition is equivalent to saying that $f\in \img(\iota_{m,n})$ (see Remark \ref{r:twass}), so we can rewrite
$$ \Hom_{\Tw(\A ss)} (\mu_m , \mu_n) = \underset{f \in \img(\iota_{m,n})}{\bigsqcup} \A_{f} .$$
As in Remark \ref{r:twass}, we write $\img(\iota_{m,n}) = \{\const_0\} \; \sqcup \; \img^{\circ}(\iota_{m,n})$. Unwinding computation, we have that

- when $f\in \img^{\circ}(\iota_{m,n})$, there is also a unique choice of $\alpha_0$ solving the equation \eqref{eq:alpha}, and

- when $f=\const_0$, there are $(n+1)$ choices of $\alpha_0\in \Sigma_{n+1}$ solving \eqref{eq:alpha} precisely given as
$$\alpha_0^{i}:=[i,1,\cdots,i-1,i+1,\cdots,n+1] \;\; , \;\; i=1,\cdots,n+1 .$$
So we can rewrite once more
$$ \Hom_{\Tw(\A ss)} (\mu_m , \mu_n) = \{ \alpha_0^{i} \}_{i=1}^{n+1} \; \sqcup \; \img^{\circ}(\iota_{m,n}) .$$
Finally, we find the desired natural bijection $\varphi_{m,n}$ separated into two components as follows:
$$ \varphi_{m,n}= \varphi_{m,n}^{\const} \, \sqcup \, \iota^{\circ}_{m,n} :  \Hom^{\const}_{\Delta}([m] , [n]) \; \sqcup \; \Hom^\circ_{\Delta}([m] , [n]) \lrarsimeq \{ \alpha_0^{i} \}_{i=1}^{n+1} \; \sqcup \; \img^{\circ}(\iota_{m,n}) $$
in which $\varphi_{m,n}^{\const}$ sends each constant map $[m] \lrar [n]$ with value $i$ to $\alpha_0^{i+1}$, while $\iota^{\circ}_{m,n}$ is the natural bijection mentioned in Remark \ref{r:twass}.
\end{proof}

\smallskip

\subsection{Main statements}\label{sub:cotantsplcop}

\smallskip

Let $\P$ be a fibrant and $\Sigma$-cofibrant $C$-colored simplicial operad. We shall now prove that the cotangent complex $\rL_\P \in \T_\P\Op(\Set_\Delta)$ can be represented as a spectrum valued functor on $\Tw(\P)$. Our treatment is inspired by the work of [[9], \S 3.3].

As introduced in the previous subsection, $\Tw(\P)$ is defined to be the image of $\P$ through the unstraightening functor $\Un_{\textbf{Ib}^{\mathcal{P}}} : \IbMod(\P) = \Fun(\textbf{Ib}^{\mathcal{P}},\Set_\Delta) \lrarsimeq  (\Set_\Delta^{\cov})_{/\sN(\textbf{Ib}^{\mathcal{P}})}$. To avoid complication of notation, we will write $\U n$ instead $\Un_{\textbf{Ib}^{\mathcal{P}}}$. Recall by construction that there is a canonical left fibration $\Tw(\P) = \U n(\P) \lrar \sN(\textbf{Ib}^{\mathcal{P}})$.

Observe first that $\U n$ induces a right Quillen equivalence (denoted by) $$\U n_{\P//\P} : \IbMod(\P)_{\P//\P} \lrarsimeq  (\Set_\Delta^{\cov})_{\Tw(\P)//\Tw(\P)}$$ where $(\Set_\Delta^{\cov})_{\Tw(\P)//\Tw(\P)}$ refers to the \textbf{pointed model category} associated to $(\Set_\Delta^{\cov})_{/\Tw(\P)}$. This induces a right Quillen equivalence of stabilizations
$$ \U n^{\Sp}_{\P//\P} : \T_\P\IbMod(\P) \lrarsimeq \Sp((\Set_\Delta^{\cov})_{\Tw(\P)//\Tw(\P)}) .$$

Observe now that the straightening functor $\St_{\Tw(\P)} : (\Set_\Delta^{\cov})_{/\Tw(\P)} \lrarsimeq \Fun(\mathfrak{C}[\Tw(\P)] , \Set_\Delta)$ lifts to a left Quillen equivalence between the associated pointed model categories
\begin{equation}\label{eq:St_P}
(\St_{\Tw(\P)})_* : (\Set_\Delta^{\cov})_{\Tw(\P)//\Tw(\P)} \lrarsimeq \Fun(\mathfrak{C}[\Tw(\P)] , (\Set_\Delta)_*)
\end{equation}
where $(\Set_\Delta)_*$ denotes the pointed model category associated to $\Set_\Delta$. The latter induces a left Quillen equivalence of stabilizations
$$ \St^{\Sp}_{\Tw(\P)} : \Sp((\Set_\Delta^{\cov})_{\Tw(\P)//\Tw(\P)}) \lrarsimeq \Fun(\mathfrak{C}[\Tw(\P)] , \Spectra) $$
where $\Spectra$ refers to the stable model category of spectra.

We now obtain a sequence of right or left Quillen equivalences
\begin{equation}\label{eq:seqeqvl}
\T_\P\Op(\Set_\Delta) \x{\U^{ib}_\P}{\lrarsimequ} \T_\P\IbMod(\P) \underset{\simeq}{\x{\U n^{\Sp}_{\P//\P}}{\xrightarrow{\hspace*{1cm}}}} \Sp((\Set_\Delta^{\cov})_{\Tw(\P)//\Tw(\P)})  \underset{\simeq}{\x{\St^{\Sp}_{\Tw(\P)}}{\xrightarrow{\hspace*{1cm}}}} \Fun(\mathfrak{C}[\Tw(\P)] , \Spectra)
\end{equation}

Let us describe the derived image of $\rL_\P\in \T_\P\Op(\Set_\Delta)$ through this composition. We have seen that $\RR\U^{ib}_\P(\rL_\P) [1] \simeq \widetilde{\rL}_\P$ (cf. Corollary \ref{conclusionCotangentCplx}). Namely, $\widetilde{\rL}_\P\in\T_\P\IbMod(\P)$ is the prespectrum with $(\widetilde{\rL}_\P)_{k,k}=\P\circ\sS_C^{k}$ where $\sS^{k}_C$ is the $C$-collection with $\sS_C^{k}(c;c)=\sS^{k}$ for every color $c$ and agreeing with $\emptyset$ on the other levels. In our setting, we will refer to $\sS^{k}$ as a Kan replacement of the (pointed) $k$-sphere so that $\P\circ\sS_C^{k}$ is fibrant. Furthermore, the right derived functor $\RR\U n^{\Sp}_{\P//\P}$ simply sends $\widetilde{\rL}_\P$ to the prespectrum $$\U n(\P\circ \sS_C^{\bullet}) \in \Sp((\Set_\Delta^{\cov})_{\Tw(\P)//\Tw(\P)})$$ with $\U n(\P\circ \sS_C^{\bullet})_{k,k} = \U n(\P\circ \sS_C^{k})$. Finally, let us denote by
\begin{equation}\label{eq:F_P}
\F_\P := \LL\St^{\Sp}_{\Tw(\P)}(\U n(\P\circ \sS_C^{\bullet})) \in \Fun(\mathfrak{C}[\Tw(\P)] , \Spectra)
\end{equation}
the derived image of $\U n(\P\circ \sS_C^{\bullet})$ through $\St^{\Sp}_{\Tw(\P)}$. So $\F_\P$ is the derived image of $\rL_\P [1]$ under the composed functor \eqref{eq:seqeqvl}.

To get a description of the functor $\F_\P$, observe first that there is an equivalence of $\infty$-categories
$$ \Fun(\mathfrak{C}[\Tw(\P)] , \Spectra)_\infty \lrarsimeq \Fun(\Tw(\P) , \textbf{Spectra}) $$
where $\textbf{Spectra}$ is the $\infty$-category of spectra. So we will regard $\F_\P$ as an $\infty$-functor $\Tw(\P) \lrar \textbf{Spectra}$. Let $\mu\in \P(c_1,\cdots,c_m;c)$ be an operation of $\P$, regarded as an object of $\Tw(\P)$. By construction, $\F_\P(\mu) \in \textbf{Spectra}$ is the prespectrum with $\F_\P(\mu)_{k,k}$ being given by the value at $\mu$ of the functor
$$\St_{\Tw(\P)}(\U n(\P\circ \sS_C^{k})) : \mathfrak{C}[\Tw(\P)] \lrar \Set_\Delta.$$ By the fact that the map $\U n(\P\circ \sS_C^{k}) \lrar \U n(\P) = \Tw(\P)$ is a left fibration and by using [[4], 2.2.3.15] (with the opposite convention), we have that
$$ \F_\P(\mu)^{\op}_{k,k} \simeq \U n(\P\circ \sS_C^{k}) \times_{\U n(\P)} \{\mu\} .$$
We have on each level that $(\P\circ \sS_C^{k}) (c_1,\cdots,c_m;c) = \P(c_1,\cdots,c_m;c) \times (\sS^{k})^{\times m}$.
With a help of Remark \ref{r:fiberkan}, it can then be computed that $\F_\P(\mu)_{k,k} \simeq (\sS^{k})^{\times m}$ and hence, we find that $\F_\P(\mu) \simeq \mathbb{S}^{\times m}$ the $m$-fold product of the \textbf{sphere spectrum}.

Let $\nu \in \P(d_1,\cdots,d_n;d)$ be another operation of $\P$ and $\widetilde{f} : \mu \lrar \nu$ a morphism in $\Tw(\P)$ lying above a map $f : \langle n \rangle \lrar \langle m \rangle$ in $\Fin_*$. Then the structure map
\begin{equation}\label{eq:spheremap}
\F_\P(\widetilde{f}) : \F_\P(\mu) = \mathbb{S}^{\times \{1,\cdots,m\}} \lrar \mathbb{S}^{\times \{1,\cdots,n\}} = \F_\P(\nu)
\end{equation}
is defined by, for each $i\in \{1,\cdots,m\}$, copying the $i$'th factor to the factors of position $j\in f^{-1}(i)$ when this fiber is nonempty or collapsing that factor to the zero spectrum otherwise.

\smallskip

We summarize the above steps in the following:
\begin{thm}\label{c:main} Let $\P$ be a fibrant and $\Sigma$-cofibrant simplicial operad. There is an equivalence of $\infty$-categories
$$ \T_\P\Op(\Set_\Delta)_\infty \lrarsimeq \Fun(\Tw(\P) , \textbf{Spectra}) .$$
Moreover, under this equivalence, the cotangent complex $\rL_\P$ is identified to $\F_\P[-1]$ the desuspension of the functor $ \F_\P : \Tw(\P)  \lrar \textbf{Spectra}$ \eqref{eq:F_P}, which is given on objects by sending each operation $\mu\in\P$ of arity $m$  to $\F_\P(\mu) = \mathbb{S}^{\times m}$. Consequently, for a given functor $\F : \Tw(\P) \lrar \textbf{Spectra}$, the $n$'th Quillen cohomology group of $\P$ with coefficients in $\F$ is computed by
$$ \sH^{n}_Q (\P ; \F) = \pi_{0}\Map_{\Fun(\Tw(\P) , \textbf{Spectra})} (\F_\P , \F[n+1]) .$$
\end{thm}

\begin{example}\label{ex:quillencom} By Proposition \ref{p:Twisgood}, the twisted arrow $\infty$-category of the little $\infty$-cubes operad $\E_\infty$ is equivalent to that of $\C om$, and hence is equivalent to $\Fin_*^{\op}$. So the tangent category $\T_{\E_\infty}\Op(\Set_\Delta)$ is (up to a zig-zag of Quillen equivalences) equivalent to $\Fun(\Fin_*^{\op},\Spectra)$ endowed with the projective model structure. The functor $\F_{\E_\infty} : \Fin_*^{\op} \lrar \Spectra$ takes each object $\l m \r$ to $\mathbb{S}^{\times m}$ and for a given map $f : \l m \r \lrar \l n \r$ in $\Fin_*$, the structure map $\F_{\E_\infty}(f) : \mathbb{S}^{\times n} \lrar \mathbb{S}^{\times m}$ is the same as the map \eqref{eq:spheremap}.
\end{example}

An immediate consequence of this example is that the stabilization of $\infty$-operads is equivalent to $\Fun(\Fin_*^{\op},\Spectra)$. For more illustration, we shall now relate Quillen cohomology of $\E_\infty$ to the one called \textbf{stable cohomotopy of right $\Gamma$-modules} (cf. [23, 24]).

Let $\textbf{k}$ be a field. Recall that a \textbf{right $\Gamma$-module} is by definition a functor $\Fin_*^{\op} \lrar \Mod_\textbf{k}$. One particularly regards the right $\Gamma$-module $t : \Fin_*^{\op} \lrar \Mod_\textbf{k}$ given by taking $\l m \r$ to $t(\l m \r)=[\l m \r,\textbf{k}]$ the $\textbf{k}$-module of based maps from $\l m \r$ to $\textbf{k}$ (where $\textbf{k}$ has basepoint $0_\textbf{k}$).

\begin{rem}\label{r:tandfcinfty}
The $\textbf{k}$-module $t(\l m \r)$ admits a canonical basis given by $\{\tau_i\}_{i=1}^{m}$ in which $\tau_i : \l m \r\lrar \textbf{k}$ is defined by sending $i$ to $1_\textbf{k}$ and sending the others to $0_\textbf{k}$. In particular, we have that
$$ t(\l m \r)=[\l m \r,\textbf{k}] \cong \textbf{k}^{\oplus \, m} \, .$$
For a given map $f : \l m \r \lrar \l n \r$ in $\Fin_*$, under the above identification, the structure map $$ t(f) :  t(\l n \r) = \textbf{k}^{\oplus \, n} \lrar \textbf{k}^{\oplus \, m} = t(\l m \r)$$
is given by, for each $i\in \{1,\cdots,n\}$, copying the $i$'th factor to the factors of position $j\in f^{-1}(i)$ when this fiber is nonempty or collapsing that factor to the zero module otherwise (i.e., in the same fashion as the map $\F_{\E_\infty}(f)$ of Example \ref{ex:quillencom}).
\end{rem}

We now fix $T : \Fin_*^{\op} \lrar \Mod_\textbf{k}$ to be a right $\Gamma$-module. According to [[24], $\S$3.4], the \textbf{$k$'th stable cohomotopy group} of $T$ is formulated as
$$ \pi^{k}T \cong \Ext_{\Gamma}^{k}(t,T) \cong \pi_0\Map^{\h}_{\Fun(\Fin_*^{\op},\C(\textbf{k}))}(t,T[k]) .$$
Here we identified right $\Gamma$-modules with functors $\Fin_*^{\op} \lrar \C(\textbf{k})$ from $\Fin_*^{\op}$ to dg $\textbf{k}$-modules, via the embedding functor $\Mod_\textbf{k} \lrar \C(\textbf{k})$. Besides that, $T[k] : \Fin_*^{\op} \lrar \C(\textbf{k})$ is the functor with $T[k](\l m \r) = (T(\l m \r)) [k]$ the $k$-suspension of $T(\l m \r)$ regarded as a chain complex concentrated in degree 0.

On other hand, the right $\Gamma$-module $T$ corresponds to a functor $\ovl{T} : \Fin_*^{\op} \lrar \Sp(\sMod_\textbf{k})$ given by the composition
$$ \ovl{T} : \Fin_*^{\op} \x{T}{\lrar}  \Mod_\textbf{k} \lrar \sMod_\textbf{k} \x{\ovl{\Sigma^\infty}}{\lrar} \Sp(\sMod_\textbf{k}) $$
in which the second functor is the obvious embedding and $\ovl{\Sigma^\infty}$ is the suspension spectrum replacement functor (cf. [8], $\S$2.3). Here we use $\ovl{\Sigma^\infty}$ (instead of $\Sigma^\infty$) to get that $\ovl{T}$ is a diagram of suspension spectra, and hence a diagram of $\Omega$-spectra. Furthermore, consider the adjunction
$$ \Sp(\mathbb{F}_*) : \adjunction*{}{\Spectra = \Sp((\Set_\Delta)_*)}{\Sp(\sMod_\textbf{k})}{} : \Sp(\mathbb{U}_*)$$
induced by the free-forgetful adjunction $\mathbb{F}_* : \adjunction*{}{(\Set_\Delta)_*}{(\sMod_\textbf{k})_* = \sMod_\textbf{k} }{} : \mathbb{U}_*$ between pointed categories. We denote by $\widetilde{T}$ the composed functor
$$ \widetilde{T} : \Fin_*^{\op} \x{\ovl{T}}{\lrar} \Sp(\sMod_\textbf{k}) \x{\Sp(\mathbb{U}_*)}{\xrightarrow{\hspace*{0.8cm}}} \Spectra .$$

\begin{prop}\label{p:stablecohomotopy} The $(k-1)$'th Quillen cohomology group of the little $\infty$-cubes operad $\E_\infty$ with coefficients in $\widetilde{T}$ is isomorphic to the $k$'th stable cohomotopy group of $T$:
$$ \sH^{k-1}_Q (\E_\infty ; \widetilde{T}) \cong \pi^{k}T .$$

\end{prop}
\begin{proof} By Theorem \ref{c:main} and Example \ref{ex:quillencom}, we just need to prove the  existence of a weak equivalence
$$ \Map^{\h}_{\Fun(\Fin_*^{\op} , \, \Spectra)} (\F_{\E_\infty} , \widetilde{T}[k]) \simeq \Map^{\h}_{\Fun(\Fin_*^{\op},\C(\textbf{k}))}(t,T[k]) .$$
By the induced Quillen adjunction $\adjunction*{}{\Fun(\Fin_*^{\op} , \Spectra)}{\Fun(\Fin_*^{\op} , \Sp(\sMod_\textbf{k}))}{}$, we have that
$$ \Map^{\h}_{\Fun(\Fin_*^{\op} , \, \Spectra)} (\F_{\E_\infty} , \widetilde{T}[k]) \simeq \Map^{\h}_{\Fun(\Fin_*^{\op} , \, \Sp(\sMod_\textbf{k}))}(\Sp(\mathbb{F}_*)\circ \F_{\E_\infty} , \ovl{T}[k] ) .$$
Here we note that $\Sp(\mathbb{F}_*)\circ\F_{\E_\infty}$ has already the right type, by Observation \ref{ob:stablecohomotopy}. By construction, for each $\l m \r \in \Fin_*$, $\Sp(\mathbb{F}_*)\circ\F_{\E_\infty}(\l m \r) = \Sp(\mathbb{F}_*)(\mathbb{S}^{\times m}) \in \Sp(\sMod_\textbf{k})$ is the prespectrum with
\begin{equation}\label{eq:stablecohomotopy}
\Sp(\mathbb{F}_*)(\mathbb{S}^{\times m})_{n,n} = \coker \, (\textbf{k} \lrar \textbf{k}\{\sS^n\}^{\otimes \, m}) \in \sMod_\textbf{k} .
\end{equation}

On other hand, consider the sequence of right or left Quillen equivalences
$$ \Sp(\sMod_\textbf{k}) \x{\Sp(\sN)}{\underset{\simeq}{\xrightarrow{\hspace*{0.8cm}}}} \Sp(\C_{\geqslant0}(\textbf{k})) \lrarsimeq \Sp(\C(\textbf{k})) \x{\Omega^\infty}{\lrarsimequ} \C(\textbf{k}) $$
where the first arrow is the stabilization of the normalized complex functor $\sN$ and the second arrow is the embedding functor. This induces a sequence of right or left Quillen equivalences
$$ \Fun(\Fin_*^{\op} , \, \Sp(\sMod_\textbf{k})) \lrarsimeq \Fun(\Fin_*^{\op} , \, \Sp(\C_{\geqslant0}(\textbf{k}))) \lrarsimeq \Fun(\Fin_*^{\op} , \, \Sp(\C(\textbf{k}))) \lrarsimeq \Fun(\Fin_*^{\op} , \, \C(\textbf{k})) . $$
Note that the derived image of $\ovl{T}[k]$ in $\Fun(\Fin_*^{\op},\C(\textbf{k}))$ is exactly $T[k]$. Combined with the first paragraph, it remains to show that the derived image of $\Sp(\mathbb{F}_*)\circ \F_{\E_\infty}$ in $\Fun(\Fin_*^{\op},\C(\textbf{k}))$ is  weakly equivalent to $t$.