Suppose the copresheaf monad $\bbP^\dag$ distributes over $\bbT$. Then the category $\TCL$ with complete and continuous   $\bbT$-algebras as objects and right adjoint $\bbT$-homomorphisms as morphisms is monadic over $\QOrd$. In particular, the category $\QCD$, of which the objects are real-enriched completely distributive lattices and the morphisms are functors that are at the time left adjoints and right adjoints, is the Eilenberg-Moore algebra category for the composite monad $\bbP\circ\bbP^\dag$ in $\QOrd$, hence monadic over $\QOrd$.

Is the full subcategory of $\TAlg$ composed of complete and continuous $\bbT$-algebras   monoidal closed?

§ REAL-ENRICHED DOMAINS

For each real-enriched category $X$, let
$$\CJ X=\{\phi\in\CI X\mid \phi ~\text{has a colimit}\}.$$
Then, every forward Cauchy net of $X$ has a Yoneda limit if and only if $\CJ X=\CI X$.

The Yoneda embedding $\sy\colon X\lra\CP X$ factors through $\CJ X$, so we'll write $\sy\colon X\lra\CJ X$ for  the Yoneda embedding with codomain restricted to $\CJ X$. It is clear that $\sy\colon X\lra\CJ X$ has a left adjoint $\colim\colon \CJ X\lra X$ that sends each $\phi\in\CJ X$ to its colimit.

(Waszkiewicz [114])  Suppose $X$ is a real-enriched category. If   $\colim\colon \CJ X\lra X$ has a left adjoint, then we say that $X$  is continuous. In other words, $X$  is continuous if there exists a string of adjunctions
\[\wayb\dashv\colim \dashv \sy\colon X\lra\CJ X.\]
A Yoneda comlete and continuous real-enriched category is called a real-enriched domain.

A real-enriched category $A$ is a real-enriched domain if and only if it is a continuous $\bbI$-algebra, where   $\bbI=(\CI,{\sf m},\sy)$ is the ideal monad in $\QOrd$; that means, $A$ is a separated real-enriched category  such that there is a string of adjunctions:
\[\wayb\dashv\colim \dashv \sy\colon A\lra\CI A.\]

For each ordered set $(X,\leq)$, it is readily verified that $\omega(X,\leq)$ is a real-enriched domain if and only if $(X,\leq)$ is a domain. But, the underlying ordered set of a real-enriched domain may fail to be a domain.

(Yu and Zhang [119])
In this example we assume that the implication  operator $\ra\colon[ 0,1]^2\lra[0,1]$ is  continuous  at every point off the diagonal. Consider the real-enriched category $X$, where $X=[0,1]\cup\{\infty\}$ and
$$ X(x,y)= \begin{cases}x\ra y & x,y\in [0,1], \\ 1 & x=y=\infty,\\  0 & x\in [0,1], y=\infty, \\  y & x=\infty,y\in[0,1]. \end{cases} $$

Since forward Cauchy nets in $X$ are  forward Cauchy nets in $([0,1],\alpha_L)$ plus constant nets with value $\infty$,   $X$ is Yoneda complete. Since  $\ra\colon[ 0,1]^2\lra[0,1]$ is  continuous  at every point off the diagonal, a routine calculation shows that
\[{\sf d}\colon X\lra \CI X, \quad {\sf d}(x)=\begin{cases}X(-,\infty) & x=\infty,\\    X(-,0) & x=0,\\ \sup\limits_{y<x}X(-,y) & x\in(0,1]\end{cases}\]
is left adjoint to $\colim\colon\CI X\lra X$, so $X$ is a real-enriched domain. But,  $X_0$ is not a domain.

Let $X$ be  a real-enriched category. The way below distributor  of $X$ refers to the distributor $\mathfrak{w}\colon X\oto X$ given by
\[\mathfrak{w}(y,x)=\inf_{\phi\in\CJ X}(X(x,\colim\phi)\ra\phi(y)).\]
It is clear that $\mathfrak{w}(y,x)\leq X(y,x)$.

( Waszkiewicz [114]) A real-enriched category  $X$ is continuous   if and only if for all $x\in X$, the weight $\mathfrak{w} (-,x)$ belongs to $\CJ X$ with $x$ being a  colimit. In this case,    $\wayb x=\mathfrak{w} (-,x)$.

We prove the necessity first. Suppose $X$ is continuous. We show that  for each $x\in X$, $\mathfrak{w} (-,x)=\wayb x$,   hence $\mathfrak{w} (-,x)$ belongs to $\CJ X$ and has $x$ as  colimit.   Since $\colim\wayb x=x$, then
$$\mathfrak{w} (-,x)\leq X(x,\colim\wayb x)\ra\wayb x= \wayb x.$$
On the other hand, since  for all   $\phi\in\CJ X$, we have
$$X(x,\colim\phi)\with\wayb x = \CJ X(\wayb x,\phi)\with\wayb x\leq \phi,$$
then $\wayb x\leq X(x,\colim\phi)\ra\phi$ for all $\phi\in\CJ X$, hence $\wayb x\leq \mathfrak{w} (-,x)$.

For sufficiency we show that the assignment $x\mapsto \wayb x\coloneqq \mathfrak{w} (-,x)$ defines a  left  adjoint of $\colim\colon\CJ X\lra X$. Let $x\in X$ and $\phi\in\CJ X$. By definition   we have $\wayb x\leq   X(x,\colim\phi)\ra\phi$, hence
$$X(x,\colim\phi)\leq \CJ X(\wayb x,\phi).$$
Since taking colimit is functorial $\CJ X\lra X$ and $\colim\wayb x=x$, then
$$\CJ X(\wayb x,\phi)\leq X(\colim\wayb x,\colim\phi)=X(x,\colim\phi).$$
Therefore, $\wayb$ is a  left adjoint  of $\colim\colon\CJ X\lra X$, as desired.

(Kostanek and Waszkiewicz [62]) The way-below distributor $\mathfrak{w}$ of a continuous real-enriched category   interpolates in the sense that $\mathfrak{w} \circ \mathfrak{w} =\mathfrak{w} $.

By Proposition <ref>, $\mathfrak{w}(-,x)$ is the least element of $\CJ X$ having $x$ as colimit. Since $\mathfrak{w}(y,x)\leq X(y,x)$ for all $y,x\in X$, it follows that $\mathfrak{w} \circ \mathfrak{w}(-,x)\leq\mathfrak{w}(-,x)$, so  we  only need to show that $\mathfrak{w}\circ\mathfrak{w}(-,x)$ belongs to $\CI X$ and has $x$ as colimit.

Step 1. $\mathfrak{w}\circ\mathfrak{w}(-,x)\in \CI X$.

Let $\lam,\mu\in\CP X$. Since
$$\sub_X(\mathfrak{w}(-,y),\lam)\with X(z,y)\with \mathfrak{w}(-,z)
\leq \lam, $$
it follows that
$$\sub_X(\mathfrak{w}(-,y),\lam)\with X(z,y)\leq \sub_X(\mathfrak{w}(-,z),\lam), $$
so the assignment $y\mapsto \sub_X(\mathfrak{w}(-,y),\lam)$ defines a weight of $X$. Then,
\begin{align*} &\quad\quad ~  \sub_X(\mathfrak{w}\circ\mathfrak{w}(-,x),\lam\vee\mu)\\ & =\bw_{y\in X}\sub_X(\mathfrak{w}(y,x)\with\mathfrak{w}(-,y),\lam\vee \mu)\\ &= \bw_{y\in X}\Big(\mathfrak{w}(y,x)\ra\sub_X(\mathfrak{w}(-,y),\lam\vee \mu)\Big)\\ &=\bw_{y\in X}\Big(\mathfrak{w}(y,x)\ra(\sub_X(\mathfrak{w}(-,y),\lam)\vee\sub_X(\mathfrak{w}(-,y),\mu)\Big)\\ &=\bw_{y\in X}\Big(\mathfrak{w}(y,x)\ra\sub_X(\mathfrak{w}(-,y),\lam)\Big)\vee\bw_{y\in X}\Big(\mathfrak{w}(y,x)\ra\sub_X(\mathfrak{w}(-,y),\mu)\Big)\\ &=\sub_X(\mathfrak{w}\circ\mathfrak{w}(-,x),\lam)\vee\sub_X(\mathfrak{w}\circ\mathfrak{w}(-,x),\mu).\end{align*}
Likewise, one verifies that for all  $\lam\in\CP X$ and  $r\in[0,1]$,
This proves that $\mathfrak{w}\circ\mathfrak{w}(-,x)\in \CI X$.

Step 2. $\colim\mathfrak{w}\circ\mathfrak{w}(-,x)=x$.

For all $b\in X$,
\begin{align*} X(x,b) &=  \bw_{y\in X}(\mathfrak{w}(y,x)\ra X(y,b)) &(\colim\mathfrak{w}(-,x)=x) \\ &= \bw_{y\in X}(\mathfrak{w}(y,x)\ra\CP X(\mathfrak{w}(-,y),X(-,b))) &(\colim\mathfrak{w}(-,y)=y)\\ &= \CP X(\colim\mathfrak{w}\circ\mathfrak{w}(-,x),X(-,b)) , \end{align*}
hence $\colim\mathfrak{w}\circ\mathfrak{w}(-,x)=x$.

The following definition is a direct extension of that for generalized metric spaces, see e.g. Bonsangue,   van Breugel and Rutten [17], and Goubaul-Larrecq [36].

An element $a$ of a real-enriched category $X$  is compact if the functor
$$X(a,-)\colon X\lra\sV$$
is Yoneda continuous, where $\sV=([0,1],\alpha_L)$.

It follows from Proposition <ref> that a Cauchy weight  of a real-enriched category $X$ is precisely  an ideal of $X$ that is a compact element of $\CP X$.

For each real-enriched category $X$ and each element $a$ of $X$, the following  are equivalent:
* $a$   is  compact.
* $X(a,\colim\phi) =\phi(a)$ for all  $\phi\in\CJ X$.
* $\mathfrak{w}(-,a)=X(-,a)$.

For all $\phi\in \CJ X$, by Example <ref> the colimit of   $X(a,-)\colon X\lra\sV$ weighted by $\phi$ is equal to the composite of $X(a,-)$ and $\phi$; that is,   $\colim_\phi  X(a,-) = \phi\circ X(a,-)=\phi(a)$.  This proves $(1)\Leftrightarrow(2)$.

$(2)\Rightarrow(3)$ For each $y\in X$,
\begin{align*}\mathfrak{w}(y,a)&=\inf_{\phi\in\CJ X}(X(a,\colim\phi)\ra\phi(y))\\ & = \inf_{\phi\in\CJ X}(\phi(a)\ra\phi(y))\\ & = X(y,a).\end{align*}

$(3)\Rightarrow(2)$ Since taking colimit is functorial, then
$$\phi(a)=\CP X(X(-,a),\phi)\leq X(a,\colim\phi) $$
for all $\phi\in\CJ X$. On the other hand, since by definition
$$\mathfrak{w}(y,a)=\inf_{\phi\in\CJ X}(X(a,\colim\phi)\ra\phi(y)),$$
then $X(a,\colim\phi)\leq \mathfrak{w}(y,a)\ra \phi(y)$ for all $y\in X$ and all $\phi\in\CJ X$,  hence
$$X(a,\colim\phi)\leq \CP X(\mathfrak{w}(-,a),\phi)=\CP X(X(-,a),\phi)=\phi(a).$$
Therefore, $X(a,\colim\phi)=\phi(a)$.

A real-enriched category $X$ is  algebraic if it is Yoneda complete and  each of its elements is a Yoneda limit of a forward Cauchy net consisting of compact elements of $X$. Said differently, $X$  is algebraic if it is Yoneda complete and for each $x\in X$, there is an ideal $\phi$ of the real-enriched category $K(X)$ composed of compact elements of $X$ such that $x$ is a colimit of the inclusion functor $i\colon K(X)\lra X$ weighted by $\phi$.

This example shows the real-enriched category $\sV=([0,1],\alpha_L)$ is algebraic if and only if so is its opposite $\sV^{\rm op}$, if and only if the continuous t-norm $\with$ is Archimedean.   We distinguish three cases.

Case 1. $\&$ is isomorphic to the Łukasiewicz t-norm. Then all elements of $\sV$ and $\sV^{\rm op}$ are compact, so both $\sV$ and $\sV^{\rm op}$ are algebraic.

Case 2. $\&$ is isomorphic to the product t-norm. Then all elements of $\sV$ are compact, $0$ is the only non-compact element of $\sV^{\rm op}$, both $\sV$ and $\sV^{\rm op}$ are algebraic.

Case 3. $\&$ is not Archimedean. Let $b$ be an idempotent element other than $0$ and $1$.  For each $a\in[0,b]$  consider the ideal $\phi=\bv_{x<a}\sV(-,x)$ of $\sV$.  Since $\colim\phi =a$ and
$$\phi(a)=\bv_{x<a}(a\ra x)\leq b<1=\sV(a,\colim\phi),$$
it follows that $a$ is not compact. By help of this fact one readily verifies that $\sV$ is not algebraic. Likewise, one sees that each $a\in[0,b]$ is not compact in $\sV^{\rm op}$, so $\sV^{\rm op}$ is not algebraic.

Every  algebraic real-enriched category is a real-enriched domain.

It suffices to check that the functor $\colim\colon \CI X  \lra X$   has a left adjoint. For each  $a\in X$, take a forward Cauchy net $\{a_i\}_{i\in D}$ of compact elements of $X$ with $a$ as a Yoneda limit, let $\wayb a$ be the Yoneda limit of the forward Cauchy net $\{\sy(a_i)\}_{i\in D}$ in $\CI X$ (existence is guaranteed by Proposition <ref>). Then, for every ideal $\phi$ of $X$,
\begin{align*} \CI X(\wayb a,\phi) &=\inf_{i\in D}\sup_{j\geq i}\CI X(\sy(a_j), \phi)&(\text{$\wayb  a$ is a Yoneda limit})\\ &=\inf_{i\in D}\sup_{j\geq i}\phi(a_j)&({\rm Yoneda~lemma}) \\ &=\inf_{i\in D}\sup_{j\geq i}X(a_j, \colim\phi)&( a_j{\rm~is~compact}) \\ &=X(a,\colim\phi),&(a~ \text{is a Yoneda limit}) \end{align*}
hence $\wayb$ is a left adjoint of $\colim$.

A real-enriched category is a real-enriched domain if and only if it is a retract  of an algebraic real-enriched category in the category of $\bbI$-algebras.
This follows from Proposition <ref> and the fact that  for each real-enriched category $X$, the category $\CI X$ of ideals of $X$ is algebraic.
For a Smyth completable real-enriched category $X$, the functor $\sy\colon X\lra\CJ X$ is both left and right adjoint to  $\colim\colon \CJ X\lra X$. In particular, $X$ is continuous.

First we show that every element of $X$ is compact; that is, $X(a,\colim\phi)=\phi(a)$ for all $a\in X$ and all $\phi\in \CJ X$. Since $X$ is Smyth completable, $\phi$ is   Cauchy, hence representable because it has a colimit. This means $\phi=X(-,b)$ for some $b\in X$. Therefore,
\begin{align*} X(a,\colim\phi)   &= X(\colim X(-,a),b) \\ &=\CP X(X(-,a), X(-,b))\\ &=\CP X(X(-,a), \phi)\\ &=\phi(a).  \end{align*}

Next we show that   $\sy\colon X\lra\CJ X$   is left adjoint to $\colim\colon \CJ X\lra X$. This is easy, since $ \CJ X(\sy(a),\phi)=\phi(a)= X(a,\colim\phi)$ for all $a\in X$ and $\phi\in \CJ X$.

Every Smyth complete real-enriched category is algebraic, hence a real-enriched domain.

The equivalence between (1) and (3) in the next corollary is due to	Ali-Akbari, Honarii, Pourmahdian and Rezaii [3].
For each separated real-enriched category $X$, the following are equivalent:
* $X$ is Smyth complete.
* The functor $\sy\colon X\to\CI X$  is both  left   and  right adjoint to  the functor $\colim\colon \CI X\to X$.
* Every forward Cauchy net of $X$ has a Yoneda limit and every element of $X$ is compact.

Theorem <ref> shows that when the continuous t-norm $\&$ is   Archimedean, Yoneda complete  real-enriched categories can be characterized purely in terms of their set of formal balls. Theorem <ref> below shows that in this case, under a mild assumption, there is a parallel result for continuous real-enriched categories.

The following notion is a direct extension of that of standard quasi-metric space in  Goubault-Larrecq and Ng [37].
Suppose $\&$ is a continuous Archimedean t-norm. A real-enriched category $X$ is standard provided that for each directed subset $\{(x_i,r_i)\}_{i\in D}$ of $\mathrm{B}X$ with $\bv_{i\in D}r_i>0$,    if   $\{(x_i,r_i)\}_{i\in D}$  has a join, then so do the following subsets:

* $\{(x_i,t\with r_i)\}_{i\in D}$ for all $t>0$;
* $\{(x_i,t\ra r_i)\}_{i\in D}$ for $t=\bv_{i\in D}r_i$.

By Lemma <ref> and Lemma <ref>,   when $\&$ is a continuous Archimedean t-norm, every Yoneda complete real-enriched category is standard.   The real-enriched category in Example <ref> is not standard, otherwise, it would be Yoneda complete by Lemma <ref>.

Suppose $\&$ is a continuous Archimedean t-norm. Let $X$ be a standard  real-enriched category and let $\{(x_i,r_i)\}_{i\in D}$ be a directed subset   of $\mathrm{B}X$ with  $\bv_{i\in D}r_i>0$. If $(x,r)$ is a join of $\{(x_i,r_i)\}_{i\in D}$,  then

* $x$ is a Yoneda limit of the forward Cauchy net $\{x_i\}_{i\in D}$ and $r=\bv_{i\in D}r_i$;
* $(x,1)$ is a join of   $\{(x_i,r\ra r_i)\}_{i\in D}$;
* for each $t\in[0,1]$, $(x,t\with r)$ is a join of   $\{(x_i,t\with r_i)\}_{i\in D}$.

It suffices to show that $x$ is a Yoneda limit of the forward Cauchy net $\{x_i\}_{i\in D}$, the other conclusions will follow by Lemma <ref>.
Let $t=\bv_{i\in D}r_i$. Since $\&$ is Archimedean, $\bv_{i\in D}(t\ra r_i)=1$. Since $X$ is standard, the directed set $\{(x_i,t\ra r_i)\}_{i\in D}$ has a join, say $(b,1)$. By Lemma <ref>, $b$ is a Yoneda limit of the forward Cauchy net $\{x_i\}_{i\in D}$, then $(b, t)$ is a join of $\{(x_i,r_i)\}_{i\in D}$ by Lemma <ref>. Therefore, $r=t$ and $x$ is isomorphic to $b$, so $x$ is a Yoneda limit of  $\{x_i\}_{i\in D}$.

The following theorem   is a combination and a slight improvement of related results in  Goubault-Larrecq and Ng <cit.> and Kostanek and Waszkiewicz <cit.>. It implies that for standard quasi-metric spaces, continuity in the sense of Definition <ref> coincides with that in the sense of Goubault-Larrecq and Ng <cit.>.

Suppose $\&$ is a continuous Archimedean t-norm and $X$ is a standard real-enriched category. The following are equivalent:

* $X$ is continuous.
* The ordered set $\mathrm{B} X$ is continuous.
In this case, the way below distributor $\mathfrak{w}$ of $X$ and the way below relation $\ll$ of $\mathrm{B}X$ determine each other via $(x,r)\ll(y,s)\iff r<s\with\mathfrak{w}(x,y)$.

$(1)\Rightarrow(2)$ First we present a fact that will be used  in Step 1 below.
Suppose $X$ is a real-enriched category and $\{(x_i,r_i)\}_{i\in D}$ is  a directed subset of $\mathrm{B}X$ with $\bv_{i\in D}r_i=1$.
Let   $\phi_D=\bv_{i\in D}\bw_{j\geq i} X(-,x_j)$ and  let $\mathrm{B}\phi_D =\{(x,r)\mid \phi_D(x)>r\}$. By  Proposition <ref>, $\mathrm{B}\phi_D$ is a directed subset of $\mathrm{B}X$. We assert that for each $(x,r)\in \mathrm{B}\phi_D$, there exists $j\in D$ such that $(x,r)\sqsubseteq(x_j,r_j)$.   To see this, suppose $(x,r)\in \mathrm{B}\phi_D$. Since $\phi_D(x)=\bv_{i\in D}\bw_{j\geq i} X(x,x_j)$, there is $i\in D$ such that $r<\bw_{j\geq i} X(x,x_j)$. Since $\bv_{j\geq i}r_j=1$, there is   $j\geq i$ such that $r<r_j\with X(x,x_j)$, so $(x,r)\sqsubseteq(x_j,r_j)$.

Now we turn to the proof of the conclusion. Since $X$ is continuous, the functor $\colim\colon \CJ X\lra X$ has a left adjoint $\wayb\colon X\lra\CJ X$. For each $y\in X$, let
$$d(y)=\{(x,r)\in\mathrm{B}X\mid r<\mathfrak{w}(x,y)\}.$$
Then $d(y)$ is a directed subset of $\mathrm{B}X$ with $(y,1)$ as a join.  In the following we prove in two steps that  every element of $\mathrm{B}X$ is the join of the elements way below it, so $\mathrm{B}X$ is continuous.

Step 1. We show that  $(x,r)\ll(y,1)$ for all $(x,r)\in d(y)$, hence $(y,1)$ is the join of the elements way below it.

Let $(x,r)\in d(y)$. Suppose $D=\{(z_i,t_i)\}_{i\in D}$ is a directed subset of $\mathrm{B}X$ with a join, say $(z,t)$, larger than $(y,1)$. It is clear that $t=1$ and $X(y,z)=1$. Since $X$ is standard,  $\bv_{i\in D}t_i=1$ and $z$ is a Yoneda limit of the forward Cauchy net $\{z_i\}_{i\in D}$.    So, $z$ is a colimit of the ideal $\phi_D$ generated by   $\{z_i\}_{i\in D}$. Since $\wayb$ is left adjoint to $\colim$ we have
$$\CJ X(\wayb z,\phi_D)=X(z,\colim\phi_D)=X(z,z)=1, $$
then $\wayb z\leq \phi_D$ and   $d(z)\subseteq \mathrm{B}\phi_D$. Since $X(y,z)=1$ and $\wayb$ is a functor, then $\wayb y\leq\wayb z$. Therefore, $d(y)\subseteq d(z)\subseteq \mathrm{B}\phi_D$ and consequently,   $(x,r)\sqsubseteq(z_j,t_j)$ for some $j\in D$. This proves that $(x,r)\ll(y,1)$.

Step 2. We show that $(x,r)\ll(y,s)$ whenever $r<s\with\mathfrak{w}(x,y)$, in particular $(x,s\with r)\ll (y,s)$ for all $(x,r)\in d(y)$,   hence  by Step 1, $(y,s)$ is the join of the elements way below it.

Suppose $D=\{(z_i,t_i)\}_{i\in D}$ is a directed subset of $\mathrm{B}X$ with a join, say $(z,t)$, larger than $(y,s)$. We wish to show that $(x,r)\sqsubseteq(z_j,t_j)$ for some $j\in D$.  Since $X$ is standard, the monotone net $\{t_i\}_{i\in D}$ converges to $t$. Since $s\leq t\with X(y,z)$ and $r<s\with\mathfrak{w}(x,y)$, it follows that
$$  r< t\with X(y,z) \with\mathfrak{w}(x,y)\leq t\with \mathfrak{w}(x,z). $$
Let $\phi= \bv_{i\in D}\bw_{j\geq i} X(-,x_j)$. Since $\phi$ is an ideal of $X$ with $z$ as a colimit,   it follows that $\mathfrak{w}(-,z)\leq \phi$, then
\begin{align*}  r &< t\with \mathfrak{w}(x,z)\\ & = t\with \bv_{w\in X}(\mathfrak{w}(w,z)\with \mathfrak{w}(x,w))\\ & \leq t\with \bv_{w\in X}(\phi(w)\with \mathfrak{w}(x,w)) \\ &= t\with \bv_{w\in X}\Big[\Big(\bv_{i\in D}\bw_{j\geq i} X(w,x_j)\Big)\with \mathfrak{w}(x,w)\Big] \\ &\leq t\with \bv_{i\in D}\Big(\bv_{w\in X}\bw_{j\geq i} X(w,x_j) \with \mathfrak{w}(x,w)\Big) \\
&\leq \Big(\bv_{i\in D}t_i\Big)\with \Big(\bv_{i\in D}\bw_{j\geq i}   \mathfrak{w}(x,x_j)\Big).\end{align*}
Therefore, there is some $j\in D$ such that $r< t_j\with\mathfrak{w}(x,x_j)\leq t_j\with X(x,x_j)$, hence   $(x,r)\sqsubseteq(z_j,t_j)$.

$(2)\Rightarrow(1)$  First we present a fact that will be used  in Step 3 below.
Suppose   $X$ is a real-enriched category, $\phi\in\CJ X$, and   $(x,r)\in\mathrm{B}X$. We assert that if $(x,r)\ll(\colim\phi,1)$, then $ \phi(x)>r$.  To see this, index the directed subset $\{(y,s)\mid \phi(y)>s\}$ of $\mathrm{B}X$ as $\{(y_i,s_i)\}_{i\in D}$. Then $\bv_{i\in D}s_i=1$ and $\{y_i\}_{i\in D}$ is a forward Cauchy net with $\colim\phi$ as a Yoneda limit. By Lemma <ref> $(\colim\phi,1)$ is a join of $\{(y_i,s_i)\}_{i\in D}$, then $(x,r)\sqsubseteq(y_i,s_i)$ for some $i\in D$, hence
$$r\leq s_i\with X(x,y_i)<\phi(y_i)\with X(x,y_i)\leq \phi(x),$$
showing that $\phi(x)>r$.

Now we turn to the proof of the conclusion.
For each $y\in X$, let
$$D=\{(x,r)\mid (x,r)\ll(y,1)\}.$$
Then $D$ is a directed set of $\mathrm{B}X$ with $(y,1)$ as a join. Index $D$ by itself, i.e. $D=\{(x_i,r_i)\}_{i\in D}$. Then $\bv_{i\in D}r_i=1$ and $\{x_i\}_{i\in D}$ is a forward Cauchy net. Let
$$k(y)=\bv_{i\in D}\bw_{j\geq i}X(-,x_j).$$
Then $k(y)$ is an ideal of $X$ with $y$ as a colimit. We show that the assignment $y\mapsto k(y)$ defines a left adjoint of $\colim\colon \CJ X\lra X$, so $X$ is continuous. We do this in three steps.

Step 1. $k(y)=\bv_{i\in D}r_i\with X(-,x_i)$. This implies that if $k(y)(x)>r$ then $(x,r)\ll(y,1)$.

For any $(x_i,r_i)\sqsubseteq(x_j,r_j)$ we have $r_i \leq X(x_i,x_j)$, then $r_i\with X(-,x_i)\leq X(-,x_j)$. Therefore,
$$r_i\with X(-,x_i)\leq \bw_{j\geq i}X(-, x_j),$$
$$k(y)\geq \bv_{i\in D}r_i\with X(-,x_i).$$
To see the converse inequality, let $t<1$. Since $r_i$ tends to $1$, there exists  $i_0\in D$ such that $r_j\geq t$ whenever $j\geq i_0$. Then
$$t\with k(y)\leq \bv_{i\geq i_0}\bw_{j\geq i}r_j\with X(-,x_j) \leq \bv_{i\in D}r_i\with X(-,x_i),$$
hence $k(y)\leq \bv_{i\in D}r_i\with X(-,x_i)$ by arbitrariness of $t$.

Step 2.  If $(x,r)\ll(y,1)$, then $(x,t\with r)\ll(y,t)$ for all $t\in[0,1]$.

If $t\with r=0$ there is nothing to prove, so in the following we assume that $t\with r>0$. Index the directed set $\{(z,s)\mid (z,s)\ll(y,t)\}$ as $\{(z_i,s_i)\}_{i\in D}$. Since $\mathrm{B}X$ is continuous,  $(y,t)$ is a join of $\{(z_i,s_i)\}_{i\in D}$, then $(y,1)$ is a join of the directed set $\{(z_i,t\ra s_i)\}_{i\in D}$, so there exists   $i\in D$ such that $(x,r)\leq (z_i,t\ra s_i)$, hence  $(x,t\with r)\leq(z_i,s_i)\ll (y,t)$.

Step 3. $k\colon X\lra \CJ X$ is left adjoint to $\colim\colon\CJ X\lra X$.

For   $y_1,y_2\in X$, let $t=X(y_1,y_2)$. If $(x,r)\ll (y_1,1)$, then $(x,t\with r)\ll(y_1,t)$ by Step 2. Since $(y_1,t)\sqsubseteq(y_2,1)$, it follows that $(x,t\with r)\ll(y_2,1)$, and then,
\begin{align*} t\with k(y_1)&= \bv\{t\with r\with X(-,x)\mid (x,r)\ll(y_1,1)\}\\ &\leq \bv\{  r\with X(-,x)\mid (x,r)\ll(y_2,1)\} \\ &\leq k(y_2),\end{align*}
which shows that $k$ is a functor. By Theorem <ref>
it remains to check that for all $y\in X$ and $\phi\in\CJ X$,
$$k(y)\leq\phi\iff y\sqsubseteq\colim\phi.$$
The direction $\implies$ is obvious since $\colim$ is functorial and $\colim k(y)=y$. For the other direction it suffices to check that $k(\colim\phi)\leq\phi$. This follows from that $\phi(x)>r$ whenever $(x,r)\ll(\colim\phi,1)$.

(Goubault-Larrecq and Ng [37])  Suppose $\&$ is a continuous Archimedean t-norm. Then a Yoneda complete real-enriched category $X$   is a real-enriched domain if and only if $\mathrm{B} X$ is directed complete and continuous.

The following corollary is a direct extension of the characterization of Smyth complete quasi-metric spaces in Romaguera and Valero [98].

Suppose $\&$ is a continuous Archimedean t-norm. Then for each separated and standard real-enriched category $X$, the following are equivalent:

* $X$ is Smyth complete.
* $\mathrm{B} X$ is directed complete and continuous, and for all $x,y\in X$ and $r>0$, $(x,r)\ll(y,1)$ if and only if $r< X(x,y)$.

$(1)\Rightarrow(2)$ Since $X$ is Smyth comple,  it is a real-enriched domain with way below distributor given by $\mathfrak{w}(x,y)=X(x,y)$. Then, by Theorem <ref> and Theorem <ref>, $\mathrm{B}X$ is directed complete and continuous, and $(x,r)\ll(y,1)$ if and only if $r< X(x,y)$.

$(2)\Rightarrow(1)$ By Theorem <ref> and Theorem <ref>, $X$ is a real-enriched domain with way below distributor given by $\mathfrak{w}(x,y)=X(x,y)$, then every element of $X$ is compact and $X$ is Smyth complete.

Consider the real-enriched category $X$ in Example <ref>. Since every forward Cauchy net of $X$ is eventually constant, it follows that $X$ is Smyth complete. But, the ordered set of formal balls of $X$ fails to be directed complete. This shows that in <ref> and <ref>  the assumption  that $\&$ is Archimedean is indispensable.

5pt Real-enriched continuous lattices

A complete (hence cocomplete)  real-enriched domain is called a real-enriched continuous lattice.

For a   real-enriched category $X$, denote by  $\idl X_0$ the set of  ideals  of the ordered set $X_0$. For each ideal $I$ of $X_0$, the conical weight
\[\Lambda(I)\coloneqq\sup_{x\in I}X(-,x)\]
is an ideal of $X$. If $X$ is finitely complete and finitely cocomplete, then, with help of Proposition <ref>, one verifies that for each ideal $\phi$ of $X$,
\[\chi(\phi)\coloneqq\{x\in  X\mid\phi(x)=1\}\]
is an ideal of $X_0$.

For each  finitely complete and finitely cocomplete real-enriched category $X$, $\Lambda\colon \idl X_0 \lra(\CI X)_0$ is  left adjoint to $\chi\colon (\CI X)_0\lra \idl X_0 $.
Suppose   $I$ is an ideal of the ordered set $X_0$ and
$\phi$ is an ideal of $X$. Then

\[I\subseteq\chi(\phi)\iff\forall x\in I,\phi(x)= 1\iff
\Lambda(I)\leq\phi,\]

hence $\Lambda\dashv \chi$.

Let $X$ be a separated and cocomplete real-enriched category. Then for each weight $\phi$ of $X$, the following are equivalent:
* $\phi$ is an ideal of $X$.
* $\phi=\Lambda(I)$ for some ideal $I$ of the ordered set $X_0$.
In this case, the colimit of $\phi$ is the join of $\chi(\phi)$ in the complete lattice $X_0$.

$(1)\Rightarrow(2)$  We show that the ideal $\chi(\phi)$ satisfies the requirement; that is, $\Lambda\circ\chi(\phi)=\phi$.  Since $\Lambda$ is left adjoint to $\chi$, it follows that $\Lambda\circ\chi(\phi)\leq\phi$. For the converse inequality, pick a forward Cauchy net $\{x_i\}_{i\in D}$ of $X$ such that
\[\phi(x)=\sup_{i\in D}\inf_{j\geq i} X(x,x_j).\]
\[I  =\Big\{x\in X\mid   x\leq \inf_{j\geq i}x_j~\text{for some}~ i\in D\Big\},\]
where $\inf_{j\geq i}x_j$ denotes the meet in   $X_0$.
Then $I$ is an ideal of $X_0$ and
\[\phi =\sup_{x\in I} X(-,x),\]
so $I\subseteq \chi(\phi)$  and consequently,   $\phi\leq\Lambda\circ\chi(\phi)$.

$(2)\Rightarrow(1)$ View the ideal $I$ of $X_0$ as a forward Cauchy net of $X$, then the conclusion follows.

Finally, we check the equality the colimit of $ \phi$ is the joint of $\chi(\phi)$; that is, $\colim \phi=\sup\chi(\phi)$, where $\sup\chi(\phi)$ denotes the join of $\chi(\phi)$ in $X_0$. Since the ideal $I$ given in   $(1)\Rightarrow(2)$ is contained in $\chi(\phi)$ and
\[\colim \phi=\sup_{z\in X}\phi(z)\otimes z,\]
then $\colim \phi\geq\sup\chi(\phi)$. The converse inequality follows from that $\colim\phi= \sup I\leq\sup\chi(\phi)$.

A functor $f\colon  X\lra Y$  between cocomplete real-enriched categories  is Yoneda continuous if and only if \(f\colon  X_0\lra Y_0\)   is Scott continuous.

As another consequence of  Proposition <ref> we present a  useful characterization of ideals of a separated and cocomplete real-enriched category.

Suppose $X$ is a separated and cocomplete real-enriched category. Then, a weight $\phi$ of  $X$ is an ideal if and only if
* $\phi(x)>0$ for some $x\in X$.
* $\phi(x\vee y)=\phi(x)\wedge\phi(y)$ for all $x,y\in X$, where $x\vee y$ denotes the join in $X_0$.
* $\phi(r\otimes x)=1$ whenever $r<\phi(x)$.

We prove  the necessity first. Since $\phi$ is an ideal, it is inhabited, hence there is some $x$ for which $\phi(x)>0$.   In the following we use Proposition <ref> to check that $\phi$ satisfies both (ii) and (iii). Pick an ideal $I$ of $X_0$ such that $\phi=\Lambda(I)$. We index the elements of $I$ by itself; that means, we write an element of $I$ as $x_i$, with $i\leq j$ if   $x_i\sqsubseteq x_j$ in $X_0$. Then
\begin{align*}
\phi(x\vee y) & =\sup_{i\in I}\inf_{j\geq i}X(x\vee y,x_j)\\
& = \sup_{i\in I}\inf_{j\geq i} X(x,x_j)\wedge X(y,x_j) \\
& = \sup_{i\in I} \Big(\inf_{j\geq i} X(x,x_j)\wedge \inf_{j\geq i}X(y,x_j)\Big)\\
& = \Big(\sup_{i\in I}\inf_{j\geq i}X(x,x_j)\Big)\wedge \Big(\sup_{i\in I}\inf_{j\geq i}X(y,x_j)\Big)  \\
& =  \phi(x)\wedge\phi(y),
\end{align*}
which shows that $\phi$ satisfies (ii). For (iii), assume that $\phi(x)>r$. Then there exists  $i_0\in I$ such that $\inf_{j\geq i_0}X(x,x_j)\geq r$, hence
\begin{align*}
\phi(r\otimes x) & =\sup_{i\in I}\inf_{j\geq i}X(r\otimes x,x_j)\\ & = \sup_{i\in I}\inf_{j\geq i}(r\ra X(x,x_j)) \\ &\geq r\ra \inf_{j\geq i_0}X(x,x_j) \\ &=1.\end{align*}

Now we prove the sufficiency. Let

$$I=\{x\in X\mid\phi(x)=1\}.$$

By (i) and (iii) one sees that $I$ is not empty. By (ii) one sees that $I$ is directed. So $I$ is an ideal of $X_0$. It is clear that $\Lambda(I)\leq\phi$. To see that $\phi$ is an idea, it suffices to check that $\phi(x)\leq\Lambda (I)(x)$ for all $x\in X$. If $\phi(x)>r$,  then $\phi(r\otimes x)=1$ by (iii), hence  $r\otimes x\in I$  and consequently, $\Lambda(I)(x)\geq X(x,r\otimes x)\geq r$. Then $\phi(x)\leq\Lambda (I)(x)$ by arbitrariness of $r$.

For each complete real-enriched category $X$, the set   $\CI X$ of ideals of $X$  is closed in $[0,1]^X$ (ordered pointwise)  under   meets and directed joins.

Let $\{\phi_i\}_{i\in J}$ be a subset of $\CI X$. The meet $\bw_{i\in J}\phi_i$ clearly satisfies (i) and (ii) in Proposition <ref>. It remains to check it also satisfies the condition (iii).  If $r<\bw_{i\in J}\phi_i(x)$, then for each $i\in J$, $r<\phi_i(x)$, hence $\phi_i(r\otimes x)=1$ and consequently, $\bw_{i\in J}\phi_i(r\otimes x)=1$. Likewise,  $\CI X$ is closed in $[0,1]^X$  under directed joins.

(Lai and Zhang [74])
For each separated and complete real-enriched category $X$, the following  are equivalent:

* $X$ is an enriched continuous lattice.

* $X_0$ is a continuous  lattice and for each $x\in X$ and each ideal $\phi$ of $X$,

$$ X(x,\colim\phi)=\inf_{y\ll x}\phi(y),$$

where $\ll$ denotes the way below relation in $X_0$.

* $X_0$ is a continuous  lattice and the map
\[{\sf d}\colon X\lra \mathcal{I}X,\quad {\sf d}(x)=\sup_{y\ll x}X(-,y)\]
is a functor, 	where $\ll$ denotes the way below relation in $X_0$.

$(1)\Rightarrow(2)$  Since $\CI X$ is closed in $[0,1]^X$  under  meets and directed joins, then $(\CI X)_0$ is a continuous lattice since so is $[0,1]$. The adjunction $\wayb \dashv \colim \colon\CI X\lra X$ ensures that   $X_0$ is a retract of $(\CI X)_0$ in the category of dcpos,  so $X_0$ is a continuous lattice.
To see that

$$ X(x,\colim\phi)=\inf_{y\ll x}\phi(y)$$
for all $x\in X$ and all $\phi\in\CI X$, it suffices to show that the left adjoint $\wayb\colon A\lra\CI A$ of $\colim \colon\CI A\lra A$ is given by $\wayb x=\sup_{y\ll x}\sy(y)$, because
$$\CI X\Big(\sup_{y\ll x}\sy(y), \phi\Big)=\sub_X\Big(\sup_{y\ll x}\sy(y), \phi\Big)= \inf_{y\ll x}\phi(y). $$

Since $x$ is a colimit of the ideal $\sup_{y\ll x}\sy(y)$, then $\wayb x\leq \sup_{y\ll x}\sy(y)$.  Since $\sup\chi(\wayb x)=\colim \wayb x=x$ by Proposition <ref>, then every element   way below $x$ in $X_0$ belongs to $\chi(\wayb x)$, so $\sup_{y\ll x}\sy(y)\leq \wayb x$.

$(2)\Rightarrow(3)$ Obvious.

By virtue of Theorem <ref> we only need to show that ${\sf d}\colon X_0\lra (\mathcal{I}X)_0$ is left adjoint to $\colim\colon  (\mathcal{I}X)_0\lra X_0$. This follows directly from Proposition <ref>, which guarantees that for each $x\in X$, $\sup_{y\ll x}X(-,y)$ is the smallest ideal  of $X$ (w.r.t. underlying order) that has $x$ as a colimit.

Suppose $X$ is a real-enriched continuous  lattice. Then for all $x,y\in X$,

$$\mathfrak{w}(y,x)=\sup_{z\ll x}X(y,z)=\sup_{z\ll x}\mathfrak{w}(y,z),$$
where $\ll$ refers to the way below relation in the complete lattice $X_0$.

The first equality follows from the argument of Theorem <ref>. The second equality follows from  the first one and the interpolative property of the way below relation in $X_0$.

Suppose $X$ is a  separated and complete real-enriched category. If  $X_0$ is a continuous lattice  and  for all $p\in [0,1]$, the map
\[X_0\lra X_0, \quad x\mapsto p\multimap x\]
is Scott continuous, then $X$ is a real-enriched continuous lattice.

We show that for each $x\in X$ and each ideal $\phi$ of $X$,
\[X(x, \colim \phi)=\inf_{y\ll x}\phi(y),\]
where $\ll$ denotes the way below relation in $X_0$.

On the one hand, since $\{y\in X\mid y\ll x\}$ is a directed set of $X_0$ with join $x$,  the conical weight $\sup_{y\ll x}X(-,y)$ is an ideal of $X$ with $x$ as colimit, then
\[X(x, \colim \phi)\geq \CI X\Big(\sup_{y\ll x}\sy(y),\phi\Big)=\inf_{y\ll x}\phi(y).\]

On the other hand,   for all $p\in[0,1]$,

\begin{align*}
p\leq X(x, \colim \phi)\implies& x\leq p\multimap \colim  \phi\\
\implies & x\leq\sup_{d\in \chi(\phi)}(p\multimap d)
\\
\implies & \forall y\ll x, ~ y\leq p\multimap d ~\text{for some}~  d\in \chi(\phi)
\\
\implies & \forall y\ll x, ~p\leq X(y,d)~\text{for some}~  d\in \chi(\phi), \\
\implies & \forall y\ll x, p\leq\sup_{d\in \chi(\phi)} X(y,d)\\
\implies  & \forall {y\ll x}, p\leq \phi(y)\\
\implies & p\leq\inf_{y\ll x}\phi(y).
\end{align*}

Therefore,  $X(x,\colim \phi)\leq\inf_{y\ll x}\phi(y)$.

For each continuous t-norm $\with$,  $\sV^{\rm op}=([0,1],\alpha_R)$ is a real-enriched continuous  lattice. We use Corollary <ref> to verify the claim. First, since the underlying order of $\sV^{\rm op}$ is the opposite of the usual order  between real numbers,   $(\sV^{\rm op})_0$ is a continuous lattice. Second, since the cotensor $p\multimap x$ of $p$ with $x$ in $\sV^{\rm op}$ is given by $p\with x$, the map $p\multimap-\colon (\sV^{\rm op})_0\lra (\sV^{\rm op})_0$ is Scott continuous.

(Lai and Zhang [74])
The following are equivalent:

* Every real-enriched completely distributive lattice is a real-enriched domain.

* The real-enriched category $\sV=([0,1],\alpha_L)$ is a real-enriched domain.

* The implication operator $\ra\colon[ 0,1]^2\lra[0,1]$ is  continuous  at every point off the diagonal.

* For each $\bbP^\dag$-algebra $X$, the inclusion $\CI X\lra\CP X$ is a right adjoint.

* The copresheaf monad $\bbP^\dag=(\CPd,{\sf m}^\dag,\syd)$ distributes over the ideal monad $\bbI=(\CI,{\sf m},\sy)$.

$(1)\Rightarrow(2)$  Obvious.

$(2)\Rightarrow(3)$ By Theorem <ref>, it suffices to show that if
\[{\sf d}\colon \sV\lra \CI\sV,\quad   {\sf d}(x)(t) =\begin{cases}t\ra 0 &x=0,\\ \sup\limits_{y<x}(t\ra y) &x>0 \end{cases} \]

is a functor, then the implication operator $\ra\colon[ 0,1]^2\lra[0,1]$ is  continuous at every point off the diagonal.

Suppose on the contrary that $\ra\colon[ 0,1]^2\lra[0,1]$ is not continuous at some point off the diagonal. Then there are idempotent elements $p,q>0$  such that the restriction of $\&$ on $[p,q]$   is isomorphic to the Łukasiewicz t-norm. Pick $x\in(p,q)$. Then

\begin{align*}\CI\sV({\sf d}(x), {\sf d}(p))&= \inf_{t\in[0,1]}\Big(\sup_{y<x}(t\ra y)\rightarrow\sup_{z<p}(t\rightarrow z)\Big)\\ &\leq \sup_{y<x}(x\rightarrow y)\rightarrow\sup_{z<p}(x\rightarrow z)\\ &=q\rightarrow p   \\ & <x\rightarrow p,\end{align*}
which shows that ${\sf d}\colon \sV\lra \CI\sV$ is not a functor, a contradiction.

$(3)\Rightarrow(4)$  It suffices to show that for each $\bbP^\dag$-algebra $X$,  $\CI X$ is closed in $\CP X$ under meets and cotensors. That $\CI X$ is closed in $\CP X$ under meets is ensured by Lemma <ref>, so we only need to show that $\CI X$ is closed in $\CP X$ under cotensors.

For  $p\in[0,1]$ and $\phi\in\CI X$, set
\[D=\{d\in X\mid p\leq\phi(d)\}.\]
Then $D$ is    a directed subset of  $X_0$. First we show that
\[\{p\multimap y\mid y\in \chi(\phi)\}\subseteq D.\]
Since $\phi=\sup_{z\in\chi(\phi)}X(-,z)$, then for all $y\in \chi(\phi)$,

\begin{align*}p\ra\phi(p\multimap y)&=p\ra\sup_{z\in \chi(\phi)} X(p\multimap y,z)\\ &\geq p\ra X(p\multimap y,y)\\ &= X(p\multimap y,p\multimap y)\\ &=1,\end{align*}

which implies that $p\multimap y\in D$.

Next, let
\[\rho=\sup_{d\in D} X(-,d).\]
We wish to show that $p\ra\phi=\rho$, from which the conclusion follows since $\rho$ is an ideal of $X$ and $p\ra\phi$ is the cotensor of $p$ with $\phi$ in $\CP X$.

That $\rho\leq p\ra\phi$ is clear.   It remains to check that $p\ra\phi(x)\leq\rho(x)$ for all $x\in X$. If $p\leq\phi(x)$, then  $x\in D$ and
\[  \rho(x)=\sup_{d\in D} X(x,d)\geq X(x,  x) =1= p\ra\phi(x).\]

If  $p>\phi(x)$, then the implication operator $\ra$ is continuous at $(p,\phi(x))$, so,
\begin{align*}p\ra\phi(x)&=p\ra\sup_{y\in \chi(\phi)} X(x,y)\\ &
=\sup_{y\in \chi(\phi)}(p\ra X(x,y))
\\ &
=\sup_{y\in \chi(\phi)} X(x,p\multimap y) \\ &
\leq\sup_{d\in D} X(x,d) &(p\multimap y\in D)\\ &
\end{align*}

$(4)\Rightarrow(5)$  Proposition <ref>.

$(5)\Rightarrow(1)$ Theorem <ref>.

* Continuity of the implication operator off the diagonal is related to the equivalence of the logic formulas
$$\exists x(p\ra q(x))\quad \text{and}\quad p\ra\exists xq(x).$$
So, in many-valued logic whether complete distributivity implies continuity depends on the structure of the truth-values.
* Ideals of real-enriched categories may be thought of as analogue of ind-objects of categories (enriched over sets), however, in contrast to the fact that the category of ind-objects of a small category with finite colimits is complete and cocomplete (see e.g. Johnstone <cit.>), the real-enriched category $\mathcal{I}X$ of ideals of a complete real-enriched category $X$ may fail to be complete. Actually, by Proposition <ref> and the above theorem, in order that $\mathcal{I}X$  be complete  for every complete real-enriched category $X$,  it is necessary and sufficient that the implication operator of the continuous t-norm $\with$ is continuous at every point off the diagonal.

\[\QCL\]
be the category composed of real-enriched continuous lattices as objects,   Yoneda continuous right adjoint functors as morphisms. This category is an analogue of the category of continuous lattices in the enriched context. When the implication operator of the t-norm $\with$ is continuous at every point off the diagonal, $\QCL$ is the Eilenberg-Moore category of the composite monad $\bbI\circ\bbP^\dag$ of the copresheaf monad and the ideal monad, hence monadic over $\QOrd$. Furthermore, in this case it is also monadic over the category of sets.

If the implication operator of  $\with$ is continuous at every point off the diagonal, then the forgetful functor \(U\colon \QCL\lra{\bf Set}\) is    monadic.

We say that a functor $k\colon X\lra X$ is a kernel operator if $k^2=k$ and $k(x)\sqsubseteq x$ in $X_0$ for all $x\in X$. If $k\colon X\lra X$ is a kernel operator, then $k\colon X\lra k(X)$ is right adjoint to the inclusion $k(X)\lra X$, so $k(X)$ is a retract of $X$ in $\QOrd$. In particular, if $X$ is a real-enriched continuous  lattice and $k\colon X\lra X$ is a Yoneda continuous kernel operator, then  $k\colon X\lra k(X)$ is a Yoneda continuous right adjoint, hence $k(X)$ is a real-enriched continuous  lattice by Corollary <ref>.

The verification of the following lemma is routine.

Suppose that $X$ is a separated and complete real-enriched category. If $R$ is an equivalence relation on   $X$ subject to the following conditions:

* $R$ is closed w.r.t. directed joins in $X_0 \times  X_0$,

* $R$ is closed w.r.t. meets in $X_0\times  X_0$,

* If $(x,y)\in R$, then  $(r\multimap x, r\multimap  y)\in R$ for all $r\in[0,1]$,
then  the map $k\colon   X\lra X$ that sends each $x\in X$ to the meet of $\{y\mid (x,y)\in R\}$ in $X_0$  is a Yoneda continuous kernel operator.

Since   the forgetful functor $U\colon \QCL\lra {\bf Set}$ is the composite of the forgetful functors $\QCL\lra \PdAlg$ and $\PdAlg\lra {\bf Set}$, it is a right adjoint. So, it suffices to check that it creates split coequalizers.

Let $f,g\colon  X\lra Y$ be a parallel pair of morphisms in $\QCL$  and let  $h\colon  Y\lra Z$ be a  split coequalizer of $f,g$ in  ${\bf Set}$. By definition there exist morphisms  $Z\to^i Y\to^j X $ in ${\bf Set}$ such that

$$h\circ f=h\circ g,~ f\circ j=\id,~ h\circ i=\id,~ g\circ j=i\circ h.$$

\[R=\{(y_1,y_2)\in Y\times Y\mid  h(y_1)=h(y_2)\}.\]
It is not hard to check that $(y_1,y_2)\in R$ if and only if there exists $(x_1,x_2)\in X\times X$ such that $g(x_1)=g(x_2)$, $y_1=f(x_1)$ and $y_2=f(x_2)$. With help of this fact, one readily verifies that $R$ satisfies the conditions (i)-(iii) in Lemma <ref>, hence  $R$ determines a Yoneda continuous kernel operator $k\colon  Y\lra  Y$. Since $k(Y)$ is a real-enriched continuous  lattice with an underlying   set   equipotent to $Z$,   $Z$ can be made into a real-enriched continuous  lattice  so that $h\colon  Y\lra Z$ is a Yoneda continuous right adjoint. This proves that  the forgetful functor $\QCL\lra {\bf Set}$ creates split coequalizers.

It is shown in Pu and Zhang [95] by a similar argument  that  for every continuous t-norm, the category $[0,1]\text{-}{\bf CDL}$ is monadic over the category of sets, where,   objects of $[0,1]\text{-}{\bf CDL}$ are real-enriched completely distributive lattices,  morphisms of $[0,1]\text{-}{\bf CDL}$ are functors  that preserve limits and colimits.

Let $F\colon{\bf Set}\lra\QCL$ be the left adjoint of $U$. In the following we present a description of the monad determined by the adjunction $F\dashv U$.

We determine the composite functor $U\circ F$ first, which turns out to be the  conical filter functor defined below.

Suppose $X$ is a real-enriched category. Following Antoniuk and Waszkiewicz [6], by a filter of $X$ we mean an ideal of its opposite $X^{\rm op}$. In other words, a filter of $X$ is a coweight $\psi$ of $X$ such that
$$\psi =\sup_{i\in D}\inf_{j\geq i} X(x_j,-) $$
for some net  $\{x_i\}_{i\in D}$  that is backward Cauchy in the sense that
$$\sup_{i\in D}\inf_{k\geq j\geq i}X(x_k,x_j)=1.$$

Suppose $X$ is a set. A  conical filter  on    $X$ is defined to be a filter of the  real-enriched category $([0,1]^X,\sub_X)$.

(Morsi [91]) Suppose $X$ is a set. Then, $\mathfrak{F}\colon [0,1]^X\lra \sQ$ is a conical filter on  $X$ if and only if  for all    $\lam,\mu\in [0,1]^X$ and all $r\in[0,1]$,
*  $\sub_X(\lam,\mu)\leq\mathfrak{F}(\lam)\ra \mathfrak{F}(\mu)$;

*  $\mathfrak{F}(1_X)=1$;
*  $\mathfrak{F}(\lam)\wedge\mathfrak{F}(\mu)=\mathfrak{F}(\lam\wedge\mu)$;
* $\mathfrak{F}(r\ra\lam)=1$ whenever $\mathfrak{F}(\lam)>r$.

Follows from Corollary <ref> directly.

Suppose $X$ is a set.
* If each member of   $\{\mathfrak{F}_i\}_{i\in J}$ is a conical  filter  on   $X$, then so is the meet $\bw_{i\in J}\mathfrak{F}_i$.

* If  $\{\mathfrak{F}_i\}_{i\in J}$ is a directed family of  conical  filters  on   $X$, then the join $\bv_{i\in J}\mathfrak{F}_i$ is a conical filter.

* If  the implication operator $\ra\colon[0,1]\times[0,1]\lra[0,1]$ is continuous at each point off the diagonal, then for each $p\in[0,1]$ and each conical filter $\mathfrak{F}$ on $X$, $p\ra\mathfrak{F}$ is a conical filter.

(i) and (ii) are a special case of Corollary <ref>; (iii) is a special case of the implication $(3)\Rightarrow(4)$ in Theorem <ref>.

For each set $X$ let ${\sf CFil}(X)$ denote the set of all conical filters on $X$. For each function $f\colon X\lra Y$ and each conical filter $\mathfrak{F}$ on $X$, the map
$$f(\mathfrak{F})\colon [0,1]^Y\lra [0,1], \quad \mu\mapsto \mathfrak{F}(\mu\circ f)$$
is readily verified to be a conical filter on $Y$.  In this way we obtain a functor
$${\sf CFil}\colon {\bf Set}\lra{\bf Set},$$
called the conical filter functor.

When the implication operator $\ra\colon[0,1]\times[0,1]\lra[0,1]$ is continuous at each point off the diagonal, the forgetful functor
has a left adjoint that maps each $\bbP^\dag$-algebra $A$ to $\CI A$. Since the left adjoint  of the forgetful functor $\PdAlg\lra {\bf Set}$ maps each set $X$ to the $\bbP^\dag$-algebra $([0,1]^X,\sub_X^{\rm op})$, we obtain the following:

Suppose the implication operator $\ra\colon[0,1]\times[0,1]\lra[0,1]$ is continuous at each point off the diagonal. Then the composite $U\circ F$ is isomorphic to  the conical filter functor CFil.

Now we determine the unit and the multiplication of the monad defined by the adjunction $F\dashv U$.
For each set $X$ and each $x\in X$, it is clear that
$${\sf e}_X(x)\colon [0,1]^X\lra[0,1], \quad \lam\mapsto \lam(x)$$
is a conical filter on $X$. The maps $\{{\sf e}_X\}_X$ constitute a natural transformation from the identity functor on Set to ${\sf CFil}$.

Suppose the implication operator $\ra\colon[0,1]\times[0,1]\lra[0,1]$ is continuous at each point off the diagonal. Then for each set $X$ and each conical filter $\wp$ on ${\sf CFil}(X)$, the map
\[\sfn_X(\wp)\colon [0,1]^X\lra [0,1],\quad \sfn_X(\wp)(\lam)=\wp(\widehat{\lam})\]
is a conical filter on $X$, where $\widehat{\lam}\colon {\sf CFil}(X)\lra\sQ$ is given by $\widehat{\lam}(\mathfrak{F})= \mathfrak{F}(\lam)$.

Since $\wp$ is a conical filter, there is a directed subset $I$ of $[0,1]^{{\sf CFil}(X)}$ (ordered pointwise) such that
$$\wp=\sup_{\xi\in I}\sub_{{\sf CFil}(X)}(\xi,-).$$
Then for each $\lam\in [0,1]^X$ it holds that
\begin{align*} \sfn_X (\wp)(\lam)&= \wp(\widehat{\lam})= \bv_{\xi\in I}\sub_{{\sf CFil}(X)}(\xi,\widehat{\lam})  = \bv_{\xi\in I} \bw_{\mathfrak{F}\in {\sf CFil}(X)}(\xi(\mathfrak{F})\ra \mathfrak{F})(\lam),\end{align*}
which implies
$$\sfn_X(\wp)=\bv_{\xi\in I} \bw_{\mathfrak{F}\in {\sf CFil}(X)}(\xi(\mathfrak{F})\ra \mathfrak{F}).$$
Therefore,  $\sfn_X(\wp)$ is a conical filter on $X$ by Proposition <ref>.

The conical filter $\sfn_X(\wp)$ is called the diagonal filter  or the Kowalsky sum of $\wp$. It is routine but a bit tedious to check that $\sfn\colon {\sf CFil}^2\lra{\sf CFil}$ is a natural transformation, and that the triple $({\sf CFil},\sfn,{\sf e})$ is the monad determined by the adjunction $F\dashv U$. We omit the verification here, instead, we display   $({\sf CFil},\sfn,{\sf e})$ as a submonad of the  double fuzzy powerset monad $\mathbb{E}=(E, \sfn,{\sf e})$.

Consider the functor $\exp\colon {\bf Set}^{\rm op}\lra{\bf Set}$ that maps each $f\colon X\lra Y$ to
$$\exp(f)\colon [0,1]^Y\lra [0,1]^X, \quad \mu\mapsto \mu\circ f.$$
It is readily verified that $\exp\colon {\bf Set}^{\rm op}\lra{\bf Set}$ is right adjoint to its opposite $\exp^{\rm op}\colon {\bf Set}\lra{\bf Set}^{\rm op}$. The monad $\mathbb{E}=(E, \sfn,{\sf e})$ defined by   $\exp^{\rm op}\dashv\exp$ is called the double fuzzy powerset monad in the category of sets. Explicitly, for each set $X$ and each function $f\colon X\lra Y$,
* $E(X)=[0,1]^{[0,1]^X}$,
* $ E(f)\colon E(X)\lra E(Y) $ is given by $E(f)(\mathfrak{A})(\mu)= \mathfrak{A}(\mu\circ f)$ for all $\mathfrak{A}\in  E(X)$ and $\mu\in[0,1]^Y$;
* the unit ${\sf e}_X\colon X\lra E(X)$ is given by ${\sf e}_X(x)(\lam)=\lam(x)$ for all $x\in X$  and $\lam\in [0,1]^X$;
* the multiplication $\sfn_X\colon E^2(X)\lra E(X)$ is given by $\sfn_X(\mathcal{H})(\lam)= \mathcal{H}(\widehat{\lam})$ for all $\lam\in [0,1]^X$ and all $\mathcal{H}\colon [0,1]^{E(X)}\lra [0,1]$, where  $\widehat{\lam}\colon E(X)\lra [0,1]$ sends  $\mathfrak{A}\in  E(X)$ to $\widehat{\lam}(\mathfrak{A})= \mathfrak{A}(\lam)$.

The conical filter functor CFil is clearly a subfunctor of $E\colon {\bf Set}\lra{\bf Set}$ through which the unit ${\sf e}$ of the monad $\mathbb{E}$ factors. If in addition the implication operator   is continuous at each point off the diagonal, then  ${\sf CFil}$ is closed with respect to the multiplication $\sfn$ of the monad $\mathbb{E}$,  hence

If   the implication operator $\ra\colon[0,1] \times[0,1]\lra[0,1]$ is continuous at each point off the diagonal, then $({\sf CFil},\sfn,{\sf e})$ is a submonad of the double fuzzy powerset monad $\mathbb{E}=(E, \sfn,{\sf e})$.

A conical filter $\mathfrak{F}$ on a set $X$ is said to be proper if $\mathfrak{F}(p_X)=p$ for all $p\in[0,1]$. Let  ${\sf PFil}(X)$ denote the set of all proper conical filters on $X$. Then we obtain a functor ${\sf PFil}\colon{\bf Set}\lra{\bf Set}$, which is a subfunctor of ${\sf CFil}$. It is not hard to check that if the implication operator $\ra\colon[0,1] \times[0,1] \lra[0,1]$ is continuous at each point off the diagonal, then ${\sf PFil}$ is a submonad of   $({\sf CFil},\sfn,{\sf e})$.   What are the Eilenberg-Moore algebras of the proper conical filter monad?

Suppose   $\bbT$ is a submonad of the presheaf monad   over which the copresheaf monad $\bbP^\dag$ distributes. By Corollary <ref>, the category $\TCL$ with complete and continuous $\bbT$-algebras as objects and right adjoint $\bbT$-homomorphisms as morphisms is monadic over $\QOrd$.  Is $\TCL$  monadic over the category of sets too?

When does the copresheaf monad $\bbP^\dag$ distributes over the the flat weight monad  or over the conically flat weight monad?

§ SATURATION

For each class of weights $\CT$, no matter $\CT$ is saturated or not, we write
\[\CT\text{-}{\sf Alg}\]
for the category of separated  $\CT$-cocomplete real-enriched categories and $\CT$-colimits preserving functors. If $\CT$ happens to be saturated, then ${\CT}\text{-}{\sf Alg}$ is  the category of Eilenberg-Moore algebras of the monad $\bbT=(\CT,{\sf m},\sy)$. In this section, we show that for each class of weights $\CT$, there is a unique saturated class of weights $\widetilde{\CT}$ such that ${\CT}\text{-}{\sf Alg}=\widetilde{\CT}\text{-}{\sf Alg}$, which implies, in particular, that ${\CT}\text{-}{\sf Alg}$ is category of Eilenberg-Moore algebras of a unique submonad of the presheaf monad.

The main result of section, namely   Theorem <ref>, is a special case of a result of Albert and Kelly [2] on saturation of class of weights in enriched category theory.

For each class of weights $\CT$, there exists a unique saturated class   of weights $\widetilde{\CT}$, called the saturation of  $\CT$, such that

* a   real-enriched category $X$  is $\CT$-cocomplete if and only if it is $\widetilde{\CT}$-cocomplete;

* a  functor   $f\colon X\lra Y$ between  $\CT$-cocomplete real-enriched categories preserves $\CT$-colimits if and only if it preserves $\widetilde{\CT}$-colimits.
In particular, ${\CT}\text{-}{\sf Alg}=\widetilde{\CT}\text{-}{\sf Alg}$.

To prove the theorem, we need to introduce two classes of weights: $\widehat{\CT} $ and $\overline{\CT} $.

For each real-enriched category $X$, define a subcategory
of $\CP X$ by putting a weight $\phi$ of $X$ in  $\widehat{\CT}X$ if, for any morphism $g\colon Y\lra Z$ of ${\CT}\text{-}{\sf Alg}$ and any functor $f\colon X\lra Y$, the colimit of $f$ weighted by $\phi$ always exists and is preserved by $g$, i.e.  $g(\colim_\phi f)=\colim_\phi(g\circ f) .$

It is readily verified that

* $\widehat{\CT}$ is a class of weights and contains the class $\CT$;

* $\widehat{\widehat{\CT}} =\widehat{\CT}$;

* $\widehat{\CT}$ is the largest class of weights for which $\widehat{\CT}\text{-}{\sf Alg}={\CT}\text{-}{\sf Alg}.$

Now we define the the class $\overline{\CT} $.
Suppose $X$ is a real-enriched category, $A$ is a subset of $\CP X$, viewed as a subcategory.   We say that $A$  is  closed under  $\CT$-colimits if, for each $\phi\in \CT A$, the colimit of the inclusion functor $i\colon A\lra\CP X$ weighted by $\phi$  belongs to $A$.
It is readily seen that if  $A\subseteq \CP X$  is closed under $\CT$-colimits, then for each functor $g\colon K\lra A$ and each $\phi\in\CT K$, the colimit of $i\circ g$ weighted by $\phi$ belongs to $A$.

For each real-enriched category $X$, let
be the intersection of all subsets of $\CP X$ that contain all representable weights of $X$ and are closed under  $\CT$-colimits. Then $\overline{\CT}X$ is the least such subset of $\CP X$. Let
* $\mathfrak{i}_X\colon \overline{\CT}X\lra\CP X$ be the inclusion functor; and
* ${\sf t}_X\colon  X\lra \overline{\CT}X$ be the Yoneda embedding with  codomain restricted to $\overline{\CT}X$.  The composite $\mathfrak{i}_X\circ {\sf t}_X$ is the Yoneda embedding $\sy_X\colon X\lra\CP X$.

For each real-enriched category $X$, $\CT X\subseteq \overline{\CT}X$.

Because for each weight $\phi$ of $X$, $\phi$ is the colimit of $\mathfrak{i}_X\circ{\sf t}_X $ weighted by $\phi$.

For each functor $f\colon X\lra Y$   and each $\phi\in \overline{\CT}X$,   $f_\exists(\phi)\in \overline{\CT}Y$.

\[Z  =
\{\phi\in\overline{\CT}X\mid f_\exists(\phi)\in\overline{\CT}Y\}.\]
It is clear that $Z$ contains all representable weights of $X$. If we can show that $Z$ is closed under  $\CT$-colimits, then $Z= \overline{\CT}X$ and the conclusion follows.

Let $j\colon  Z \lra\CP X$ be the inclusion functor. We show that for each
$\Phi\in \CT Z$, the colimit of $j$ weighted by $\Phi$ belongs to $ Z$.

Since $f_\exists\colon\CP  X\lra \CP Y$ preserves colimits, then
\begin{align*} f_\exists({\colim}_\Phi j) &= {\colim}_\Phi(f_\exists \circ j) . \end{align*}
Since $f_\exists\circ j$ factors through $\overline{\CT}Y$ by definition of $Z$, $\overline{\CT}Y$ is  closed under $\CT$-colimits, it follows that
$${\colim}_\Phi(f_\exists \circ j)\in \overline{\CT}Y,$$
which implies that ${\colim}_\Phi j \in Z$, as desired.

The above two lemmas imply that $\overline{\CT}$ is a class of weights on $\QOrd$.  The assignment $X\mapsto \overline{\CT}X$ defines a functor
$$\overline{\CT}\colon\QOrd \lra{\CT}\text{-}{\sf Alg},$$
which is indeed  left adjoint to  the forgetful functor $U\colon {\CT}\text{-}{\sf Alg} \lra \QOrd $, as we see below. This is a  special case of a general result of Kelly [58] in the theory of enriched categories.

The functor $\overline{\CT}\colon \QOrd \lra{\CT}\text{-}{\sf Alg} $ is left adjoint to the forgetful functor $U\colon {\CT}\text{-}{\sf Alg} \lra \QOrd$.

To prove Proposition <ref> we need a lemma.

For each  $\CT$-cocomplete real-enriched category $Y$,
\[\overline{\CT}Y\subseteq \{\phi\in\CP Y\mid \phi~\text{has a colimit}\}.\]

It suffices to show that

\[W \coloneqq\{\phi\in\CP Y\mid \phi~\text{has a colimit}\}\]

is closed under   $\CT$-colimits.

Let $j\colon W \lra\CP Y$ be the inclusion map. We wish to show that for each $\Phi\in\CT W$,  the colimit $\colim_\Phi j$, as a weight  of $Y$, has a colimit. First of all, by Corollary <ref> we have
\[{\colim}_\Phi j=\sup_{\phi\in W}\Phi(\phi)\with \phi.\]

Since every $\phi\in W$ has a colimit, assigning to each $\phi\in W$  its colimit defines a functor $h\colon W \lra Y$. Since $Y$ is $\CT$-cocomplete, the colimit of   $h\colon W \lra Y$ weighted by $\Phi$ exists, say $b$. We claim that $b$ is a colimit of the weight $\colim_\Phi j$ of $Y$.
For this we calculate: for all $y\in Y$,

\begin{align*}Y(b,y)&=\CP Y({\colim}_\Phi\thinspace h,\sy(y)) \\
&=\CP W(\Phi,h^{-1}(\sy(y))) \\
&= \inf_{\phi\in W} (\Phi(\phi)\ra Y(h(\phi),y) )\\
&= \inf_{\phi\in W}\Big[\Phi(\phi)\ra\Big( \inf_{z\in Y}(\phi(z)\ra Y(z,y) )\Big) \Big] \\
&= \inf_{z\in Y}\Big[\Big(\sup_{\phi\in W}\Phi(\phi)\with \phi(z)\Big) \ra Y(z,y) \Big]\\
&= \CP Y({\colim}_\Phi\thinspace j,\sy(y)),
\end{align*}
which shows that $b$ is a colimit of the weight $\colim_\Phi j$ of $Y$.

It suffices to show that for each functor $f\colon X\lra Y$ with $Y$ in ${\CT}\text{-}{\sf Alg}$, there is a unique $\CT$-colimits preserving functor  $\overline{f}\colon \overline{\CT}X\lra Y$ such that $f=\overline{f}\circ  {\sf t}_X$.

Since  the real-enriched category $W$ given in Lemma <ref> is  $\CT$-cocomplete, it is readily  verified that $\overline{\CT}X$ is contained in
\[\{\phi\in\CP X\mid f_\exists(\phi)\in W\}.\]
Let $h\colon W\lra Y$ be the functor that sends each $\phi$ of $W$ to its colimit.
\[\overline{f}\coloneqq h\circ f_\exists\colon  \overline{\CT}X\lra W \lra Y\]
satisfies the requirement.   This proves the existence.
To see the uniqueness, suppose
$g\colon \overline{\CT}X\lra Y$ satisfies the requirement. Then
\[\{\phi\in\overline{\CT}X\mid g(\phi)=  \overline{f}(\phi)\}\]
contains all representable weights of $X$ and is closed under  $\CT$-colimits. Hence,   $g=\overline{f}$ by definition of  $\overline{\CT}X$.

For each   $Y$ of ${\CT}\text{-}{\sf Alg}$, the functor  $\colim\colon \overline{\CT}Y\lra Y$   sending each $\phi$ to its colimit is the component  of the counit of the adjunction $\overline{\CT}\dashv U$ in Proposition <ref>. Hence for each  morphism  $g\colon Y\lra Z$ of ${\CT}\text{-}{\sf Alg}$, the following square is commutative:
\begin{equation*}\label{counit of S dashv U}\bfig\Square[\overline{\CT}Y`\overline{\CT}Z`Y`Z; g_\exists`\colim`\colim`g]\efig\end{equation*}
This fact will be used in the proof of Theorem <ref>.

For existence we show that $\widetilde{\CT}\coloneqq\widehat{\CT}$ satisfies the requirement.   Since  ${\CT}\text{-}{\sf Alg} =\widehat{\CT}\text{-}{\sf Alg}$, it remains to show that $\widehat{\CT}$ is saturated.

Since $\overline{\CT} X$ is closed under $\CT$-colimits in $\CP X$, if we can prove that $\overline{\CT} X=\widehat{\CT}X$  for each real-enriched category $X$,  then $\widehat{\CT}X$ is closed under $\CT$-colimits, hence under $\widehat{\CT}$-colimits, and consequently, $\widehat{\CT}$ is saturated.

First  we prove that $\overline{\CT}X\subseteq\widehat{\CT}X$. That means, if $\phi\in \overline{\CT}X$ and $g\colon Y\lra Z$ is a morphism in ${\CT}\text{-}{\sf Alg}$, then for any functor $f\colon X\lra Y$, the colimit of $f$ weighted by $\phi$   exists and is preserved by $g$. Since $f_\exists(\phi)\in \overline{\CT}Y$,   the colimit of   $f$ weighted by $\phi$ exists. By commutativity of the above  square we have $\colim(g_\exists\circ f_\exists(\phi))= g(\colim f_\exists(\phi))$,  hence the colimit of $f$ weighted by $\phi$    is preserved by $g$.

Next we prove that $\widehat{\CT}X\subseteq\overline{\CT}X$.
Since ${\CT}\text{-}{\sf Alg} =\widehat{\CT}\text{-}{\sf Alg}$, each $A$ of ${\CT}\text{-}{\sf Alg}$ is $\widehat{\CT}$-cocomplete.
Since $\overline{\CT}X$ belongs to ${\CT}\text{-}{\sf Alg}$, it is  $\widehat{\CT}$-cocomplete.  Since the inclusion   $\mathfrak{i}_X\colon \overline{\CT}X\lra\CP X$ preserves $\CT$-colimits,  it  preserves $\widehat{\CT}$-colimits too. Then, for each $\phi\in \widehat{\CT}X$,
$$ {\colim}_\phi\thinspace{\sf t}_X = {\colim}_\phi(\mathfrak{i}_X\circ{\sf t}_X)={\colim}_\phi\thinspace\sy_X=\phi,$$
hence $\phi\in \overline{\CT}X$.

To see uniqueness, suppose $\mathcal{S} $ is a saturated class of weights with $\mathcal{S}\text{-}{\sf Alg}= {\CT}\text{-}{\sf Alg}$.  Then we prove that $\mathcal{S} =\widetilde{\CT}$ by showing that $\mathcal{S}$ is a subclass of $\widehat{\CT}$ and contains the class $\overline{\CT}$.

Since $\widehat{\CT}$  is the largest class of weights for which ${\CT}\text{-}{\sf Alg} =\widehat{\CT}\text{-}{\sf Alg}$,   $\mathcal{S}$ is   a subclass of $\widehat{\CT}$.
To see that $\mathcal{S}$   contains the class $\overline{\CT}$, first  we show that $\mathcal{S}$   contains the class $\CT$; that is, $\CT X\subseteq \mathcal{S}X$ for each real-enriched category $X$. Since $\mathcal{S}$ is saturated, then
* $\mathcal{S}X$ is $\mathcal{S}$-cocomplete, hence $\CT$-cocomplete;
* the inclusion  functor $j_X\colon \mathcal{S} X \lra\CP X$ preserves $\mathcal{S} $-colimits, hence $\CT$-colimits.  Let ${\sf s}_X\colon X\lra \mathcal{S}X $ be  the Yoneda embedding with codomain restricted  to $\mathcal{S}X $.  Then for each $\phi\in \CT X$,
\[{\colim}_\phi\thinspace  {\sf s}_X = {\colim}_\phi(j_X\circ {\sf s}_X)  = {\colim}_\phi\thinspace \sy_X =\phi,\]
hence $\phi\in\mathcal{S}X$ and $\CT X\subseteq\mathcal{S}X $.

Now we show that $\mathcal{S}$   contains  $\overline{\CT}$. Since $\mathcal{S}$ is saturated and contains the class $\CT$, it follows that for each real-enriched category $X$, $\mathcal{S}X $ is closed in $\CP X$ under $\CT$-colimits, then  $\overline{\CT}X\subseteq \mathcal{S}X$.

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