given by the oplax normal natural transformation of equation~\refs{eqpointkanextensiontwocatlax} is an isomorphism of categories (notice that the $2$-naturality in $C$ and $A$ is granted, where the latter is using that $L$ is a $2$-functor).
\end{defne}

\begin{rem}
We now prove the pointwise Kan extension result for the $2$-$\Set$-enriched Grothendieck construction we were aiming at (from \remx\ref{remaimatpointwise}). Such result is original.
\end{rem}

\begin{teor}\label{teorpointkanextforgroth}
Let $F\:\A\to \CAT$ be a $2$-functor with $\A$ a small $2$-category. Then the $2$-$\Set$-enriched Grothendieck construction lax comma object square
\sq[l][7][7][\lax \opn{comma}][2.7][2.2][0.65]{\Grothop{F}}{\1}{\A}{\CAT}{}{\groth{F}}{\1}{F}
$$F=\Lan{\groth{F}}{\Delta 1}.$$
\end{teor}
\begin{proof}
By \thex\ref{grothconstrislaxcomma} (and \prox\ref{laxnatinsidegrothconstr}) we know that the lax natural transformation $\lambda$ that presents the lax comma object is lax normal.

Given $A\in \A$ and $C\in \CAT$, we prove that the oplax normal natural transformation
\begin{equation*}
\HomC{\A}{\groth{F}(-)}{A}\aRR{F}\HomC{\CAT}{(F\c \groth{F})(-)}{F(A)}\aRR[\oplaxn]{\HomC{\CAT}{\lambda_{-}}{\id{}}}\HomC{\CAT}{\Delta 1(-)}{F(A)},
\end{equation*}
that we call $\mu$, is $2$-universal. Explicitly, $\mu$ has components
\begin{fun}
\mu_{(B,X)} & \: & \HomC{\B}{B}{A} & \too & \HomC{\CAT}{1}{F(A)} \\[1ex]
&& \tc*[6]{B}{A}{u}{v}{\theta} &\mtoo & \tc*[9][30][pos=0.59][pos=0.59][,shift right=0.4ex]{1}{F(A)}{F(u)(X)}{F(v)(X)}{F(\theta)_X}
\end{fun}
for every $(B,X)\in \Grothop{F}$ and structure $2$-cells
$${\left(\mu_{(g,\gamma)}\right)}_u=F(u)(\gamma)\:F(u\c g)(X')\to F(u)(X)$$
on every $(g,\gamma)\:(B,X)\al{} (B',X')$ in $\Grothop{F}$, for every $u\:B\to A$ in $\A$. Given
$$\sigma\:\HomC{\A}{\groth{F}(-)}{A}\aoplaxn{}\HomC{\CAT}{\Delta 1(-)}{C},$$
we prove that there exists a unique functor $s\:F(A)\to C$ such that
$$(s\c -)\c \mu =\sigma.$$
We see that there is at most one such $s$, as we need, for every morphism $\alpha\:X\to X'$ in $F(A)$,
And this $s$ works thanks to the fact that $\sigma$ is oplax normal, since it is readily shown to be functorial (using that $\sigma$ is oplax) and for every $(g,\gamma)\:(B,X)\to (B',X')$ in $\Grothop{F}$ and every $\theta\:u\aR{}v\:B\to A$ in $\A$, considering $\underline{u}^X=(u,\id{})$ and $\underline{\theta}^X\:(u,F(\theta)_X)\aR{}(v,\id{})$,

We now prove the $2$-dimensional universality of $\mu$. Given $$\Xi\:\sigma\aM{}\sigma'\:\HomC{\A}{\groth{F}(-)}{A}\aoplaxn{}\HomC{\CAT}{\Delta 1(-)}{C},$$
we prove that there exists a unique natural transformation $\xi\:s\aR{} s'\:F(A)\to C$ such that
$$(\xi\ast -)\h \mu =\Xi.$$
We see that there is at most one such $\xi$, as we need, for every $X\in F(A)$,
And this $\xi$ works since it is readily shown to be natural (using that $\Xi$ is a modification) and for every $(B,X)\in \Grothop{F}$ and $u\:B\to A$ in $\A$
We have thus shown that $\mu$ is $2$-universal and this concludes the proof.
\end{proof}

\begin{rem}
The rest of this section is dedicated to the proof that every pointwise left Kan extension in $\twoCATlax$ along a $2$-$\Set$-opfibration (as defined in \defx\ref{defpointkanextensionlaxthreecat}) is a weak left Kan extension in $\twoCATlax$ as well.

For this, we need a $\oplaxn\mbox{-}\lax$ generalization of the parametrized Yoneda lemma (\thex\ref{parametrizedyonedaoplaxnlax}) that does not seem to appear in the literature. This lemma will also shed further light on the concept of oplax normal natural transformation. Indeed the idea is the following: a fully lax parametrized Yoneda lemma is not possible, since it is thanks to the strict naturality that we can classically expand the datum on the identity to a complete natural transformation, but our version shows the minimal strictness that we need in order to do so.

Interestingly, such expansion in the fully strict $2$-natural case classically depends on the naturality of what will be our parameter $A$. Instead, we will need to expand through the slight strictness of the oplax normal naturality in $B$. And an expansion through $B$ is harder to achieve than one through $A$.
\end{rem}

\begin{defne}\label{defoplaxnlax}
Let $G,H\:\B\op\times \C\to \E$ be $2$-functors. An \dfn{$\text{\textbf{oplax}}^\text{\textbf{n}}\mbox{-}\text{\textbf{lax}}$ natural transformation} $\alpha$ from $G$ to $H$ is a collection of morphisms
$$\alpha_{B,C}\:G(B,C)\to H(B,C)$$
in $\E$ for every $(B,C)\in \B\op\times\C$ and, for every $f\:B'\to B$ in $\B\op$ and $g\:C\to C'$ in $\C$, structure $2$-cells
\begin{eqD*}
\sq*[o][6][6][\alpha_{f,C}][3.1][3]{G(B',C)}{H(B',C)}{G(B,C)}{H(B,C)}{\alpha_{B',C}}{G(f,\id{})}{H(f,\id{})}{\alpha_{B,C}}\qquad
\sq*[l][6][6][\alpha_{B,g}][3.1][3]{G(B,C)}{H(B,C)}{G(B,C')}{H(B,C')}{\alpha_{B,C}}{G(\id{},g)}{H(\id{},g)}{\alpha_{B,C'}}
\end{eqD*}
such that $\alpha_{-,C}$ is oplax normal natural in $B\in \B\op$, $\alpha_{B,-}$ is lax natural in $C\in \C$ and\v[0.5]
\begin{eqD*}
\begin{cd}*[5][-3]
\&[-1ex] G(B,C)\arrow[rr,"{\alpha_{B,C}}"]\arrow[rd,"{G(\id{},g)}"{inner sep=0.2ex},""'{name=A}]\&[-1ex]\& H(B,C)\arrow[rd,"{H(\id{},g)}"{inner sep=0.2ex}]\arrow[d,Rightarrow,shift right=2.85ex,"{\alpha_{B,g}}"{pos=0.44},shorten <=0.8ex,shorten >=2.2ex]\\
G(B',C)\arrow[rd,"{G(\id{},g)}"'{inner sep=0.2ex},""{name=B}]\arrow[ru,"{G(f,\id{})}"{inner sep=0.2ex}]\&\&G(B,C')\arrow[rr,"{\alpha_{B,C'}}"]\&\vphantom{.}\arrow[d,Rightarrow,shift right=2.85ex,"{\alpha_{f,C'}}",shorten <=1.05ex,shorten >=1ex]\& H(B,C')\\
\& G(B',C')\arrow[ru,"{G(f,\id{})}"'{inner sep=0.2ex}]\arrow[rr,"{\alpha_{B',C'}}"']\&\&H(B',C')\arrow[ru,"{H(f,\id{})}"'{inner sep=0.2ex}]
\arrow[equal,from=A,to=B,shorten <=3.64ex,shorten >=3.64ex]
\end{cd}\h[2] = \h[2]
\begin{cd}*[5][-3]
\& G(B,C)\arrow[rr,"{\alpha_{B,C}}"]\arrow[d,Rightarrow,shift left=2.85ex,"{\alpha_{f,C}}"'{pos=0.44},shorten <=0.8ex,shorten >=2.2ex]\&\&[-1ex] H(B,C)\arrow[rd,"{H(\id{},g)}"{inner sep=0.2ex},""'{name=A}]\\
G(B',C)\arrow[rr,"{\alpha_{B',C}}"]\arrow[rd,"{G(\id{},g)}"'{inner sep=0.2ex}]\arrow[ru,"{G(f,\id{})}"{inner sep=0.2ex}]\&\vphantom{.}\arrow[d,Rightarrow,shift left=2.85ex,"{\alpha_{B',g}}"',shorten <=1.05ex,shorten >=1ex]\&H(B',C)\arrow[ru,"{H(f,\id{})}"{inner sep=0.2ex}]\arrow[rd,"{H(\id{},g)}"'{inner sep=0.2ex},""{name=B}]\&\&[-1ex] H(B,C')\\
\& G(B',C')\arrow[rr,"{\alpha_{B',C'}}"']\&\&H(B',C')\v[2]\arrow[ru,"{H(f,\id{})}"'{inner sep=0.2ex}]
\arrow[equal,from=A,to=B,shorten <=3.64ex,shorten >=3.64ex]
\end{cd}
\end{eqD*}
We call the last axiom the compatibility axiom.

A \dfn{modification} $\Theta\:\alpha\aM{}\beta\:G\aR[\opln\mbox{-}\lax]{}H$ between $\oplaxn\mbox{-}\lax$ natural transformations is a collection of $2$-cells
\tc+[7]{G(B,C)}{H(B,C)}{\alpha_{B,C}}{\beta_{B,C}}{\Theta_{B,C}}
in $\E$ that forms both, fixing $C$, a modification $\alpha_{-,C}\aM{}\beta_{-,C}$, and, fixing $B$, a modification $\alpha_{B,-}\aM{}\beta_{B,-}$.
\end{defne}

\begin{teor}[The $\oplaxn\mbox{-}\lax$ parametrized Yoneda lemma]\label{parametrizedyonedaoplaxnlax}
Let $K\:\B\to \A$ be a $2$-$\Set$-opfibration (with small fibres) and $F\:\B\op\times \A\to \CAT$ be a $2$-functor. There is a bijection between
$$\alpha_{B,A}\:\HomC{\A}{K(B)}{A}\to F(B,A)$$
$\oplaxn\mbox{-}\lax$ natural in $(B,A)\in \B\op\times \A$ and
$$\eta_B\:1\to F(B,K(B))$$
extraordinary lax natural in $B\in \B$ \pteor{see Hirata's~[8] for a definition of extraordinary lax natural transformations and modifications between them}.

Moreover this bijection extends to an isomorphism of categories, considering as morphisms of the two categories respectively the modifications between $\oplaxn\mbox{-}\lax$ natural transformations and the modifications between extraordinary lax natural transformations.
\end{teor}
\begin{proof}
By \thex\ref{essentialimagegivenbytwosetopf}, we can assume that $K$ is in the form $\groth{M}\:\Grothop{M}\to \A$ for a $2$-functor $M\:\A\op \to \CAT$. Given
$$\alpha_{(B,Y),A}\:\HomC{\A}{K(B,Y)}{A}\to F((B,Y),A)$$
$\oplaxn\mbox{-}\lax$ natural in $((B,Y),A)\in {\left(\Grothop{M}\right)}\op\times \A$ (see \defx\ref{defoplaxnlax}), we construct $\eta_{(B,Y)}$ as the composite
$$1\ar{\id{B}} \HomC{\A}{B}{B}\ar{\alpha_{(B,Y),B}}F((B,Y),B)$$
Then $\eta_{(B,Y)}$ is extraordinary lax natural in $(B,Y)\in \Grothop{M}$, with structure $2$-cell on $(g,\gamma)\:(B,Y)\to (B',Y')$ in $\Grothop{M}$ given by the pasting
\begin{eqD}{diagramfromalphatoeta}
\begin{cd}*[5][5]
1 \arrow[r,"{\id{B}}"]\arrow[d,"{\id{B'}}"']\& \HomC{\A}{B}{B} \arrow[ld,equal,shorten <=5.05ex,shorten >=5.05ex]\arrow[d,"{g\c -}"] \arrow[r,"{\alpha_{(B,Y),B}}"]\&F((B,Y),B) \arrow[dd,"{F(\id{},g)}"] \arrow[ld,Rightarrow,"{\alpha_{(B,Y),g}}"{pos=0.61},shorten <=4.45ex,shorten >=2.25ex]
\\
\HomC{\A}{B'}{B'}\arrow[d,"{\alpha_{(B',Y'),B'}}"'] \arrow[r,"{-\c g}"']\& \HomC{\A}{B}{B'}\arrow[rd,"{\alpha_{(B,Y),B'}}"{inner sep =0.2ex}]\arrow[ld,Rightarrow,"{\alpha_{(g,\gamma),B'}}"{pos=0.56},shorten <=3.9ex,shorten >=3.35ex]
\\
F((B',Y'),B') \arrow[rr,"{F((g,\gamma),\id{})}"']\&\&F((B,Y),B')
\end{cd}
\end{eqD}
Indeed it is true in general that, given $2$-functors $T,S\:\C\op\times \C\to \E$ the composite
is extraordinary lax in $C\in \C$ if $\iota_C$ is extraordinary lax in $C$ and $\beta_{C,D}$ is $\oplaxn\mbox{-}\lax$ in $(C,D)\in \C\op\times \C$, with structure $2$-cells given by a pasting like that of equation~\refs{diagramfromalphatoeta} (with now a possibly non-identity $2$-cell also in the upper left square). Such structure $2$-cells surely respect the identities, and they also respect the composition by the compability axiom of $\oplaxn\mbox{-}\lax$ (and oplax and lax naturality). The two dimensional axiom is satisfied as well by moving the external $2$-cell through the diagram using the three $2$-dimensional properties that we have. We can then apply this result to $\eta_{(B,Y)}$ since $\id{K(B,Y)}$ is extraordinary natural in $(B,Y)$ and $\alpha_{(B,Y),K(B',Y')}$ is $\oplaxn\mbox{-}\lax$ natural in $((B,Y),(B',Y'))$ (as $\alpha{(B,Y),A}$ is $\oplaxn\mbox{-}\lax$ in $((B,Y),A)$).\v[0.3]

Now, given $\eta_B\:1\to F((B,Y),B)$ extraordinary lax natural in $(B,Y)\in \Grothop{M}$, we expand it to functors
$$\alpha_{(B,Y),A}\:\HomC{\A}{K(B,Y)}{A}\to F((B,Y),A)$$
$\oplaxn\mbox{-}\lax$ natural in $((B,Y),A)\in {\left(\Grothop{M}\right)}\op\times \A$ as follows, using the oplax normal naturality in $(B,Y)$ (that is the only strictness we have). Given $u\:B\to A$ in $\A$, considering $\underline{u}^{Y}=(u,\id{})$, the structure $2$-cell $\alpha_{(u,\id{}),A}=\id{}$ will give us a commutative square
\sq[n][5.5][10]{\HomC{\A}{A}{A}}{F((A,M(u)(Y)),A)}{\HomC{\A}{B}{A}}{F((B,Y),A)}{\alpha_{(A,M(u)(Y)),A}}{-\c u}{F((u,\id{}),A)}{\alpha_{(B,Y),A}}
So, looking at how we constructed $\eta$ from $\alpha$, in order to reach the bijection we want, we define
$$\alpha_{(B,Y),A}(u)\deq F((u,\id{}),A)\left(\eta_{(A,M(u)(Y))}\right).$$
Given $\theta\:u\aR{} v\:B\to A$ in $\A$, considering
$$\underline{\theta}^{Y}\:(u,M(\theta)_Y)\aR{}(v,\id{})\:(B,Y)\to (A,M(v)(Y))$$
and using that $\alpha_{(v,\id{}),A}=\id{}$, we will have by the $2$-dimensional axiom of oplax normal naturality that
$$\alpha_{(B,Y),A}(\theta)=F(\underline{\theta}^Y,A)_{\h\alpha_{(A,M(v)(Y)),A}(\id{A})}\c {\left(\alpha_{(u,M(\theta)_Y),A}\right)}_{\id{A}}$$
So we firstly define the components of the structure $2$-cells that express the oplax normal naturality of $\alpha_{(B,Y),A}$ in $(B,Y)$ and then we will read how to define the action of $\alpha_{(B,Y),A}$ on morphisms $\theta$.

Looking at the diagram of equation~\refs{diagramfromalphatoeta} applied to $(\id{B},\gamma)\:(B,Y)\to (B,Y')$ in $\Grothop{M}$, we see that, in order to have a bijection between the $\alpha$'s and the $\eta$'s, we need to define
$${\left(\alpha_{(\id{B},\gamma),B}\right)}_{\id{B}}\deq \eta_{(\id{B},\gamma)}.$$
Whence, given arbitrary $(g,\gamma)\:(B,Y)\to (B',Y')$ in $\Grothop{M}$ and $u\:B\to A$ in $\A$, since
$$(u,\id{})\c (g,\gamma)=(\id{A},M(u)(\gamma))\c (u\c g,\id{}),$$ we need to define
$${\left(\alpha_{(g,\gamma),A}\right)}_u\deq F((u\c g,\id{}),A)\left(\eta_{(\id{A},M(u)(\gamma))}\right).$$
And at this point we define, by the argument above,
$$\alpha_{(B,Y),A}(\theta)\deq F(\underline{\theta}^Y,A)_{\eta_{(A,M(v)(Y))}}\c F((u,\id{}),A)\left(\eta_{(\id{A},M(\theta)_Y)}\right)$$
for every $\theta\:u\aR{}v\:B\to A$ in $\A$.

Looking at the diagram of equation~\refs{diagramfromalphatoeta} applied to $(g,\id{})\:(B,Y)\to (B',M(g)(Y))$ in $\Grothop{M}$, we see that, in order to have a bijection between the $\alpha$'s and the $\eta$'s, we need to define
$${\left(\alpha_{(B,Y),g}\right)}_{\id{B}}\deq \eta_{(g,\id{})}.$$
Whence, given an arbitrary $f\:A\to A'$ in $\A$ and $u\:K(B,Y)\to A$ in $\A$, by the compatibility axiom of $\oplaxn\mbox{-}\lax$ applied to $(u,\id{})\:(B,Y)\to (A,M(u)(Y))$ in $\Grothop{M}$ and $f\:A\to A'$ in $\A$, we need to define
$${\left(\alpha_{(B,Y),f}\right)}_u\deq F((u,\id{}),A')\left(\eta_{(f,\id{})}\right).$$

Now, we verify that such assignments work. To show that $\alpha_{(B,Y),A}$ is a functor, consider
$$u\aR{\theta} v\aR{\rho}w\:B\to A$$
in $\A$. Then
$$\alpha_{(B,Y),A}(\rho\c \theta)\deq F(\underline{(\rho\c \theta)}^Y,A)_{\eta_{(A,M(w)(Y))}}\c F((u,\id{}),A)\left(\eta_{(\id{A},M(\rho\c\theta)_Y)}\right)$$
while $\alpha_{(B,Y),A}(\rho)\c\alpha_{(B,Y),A}(\theta)$ is equal to
$$\scalebox{0.935}{$F(\underline{\rho}^Y,A)_{\eta_{(A,M(w)(Y))}}\c F((v,\id{}),A)\left(\eta_{(\id{A},M(\rho)_Y)}\right)\c F(\underline{\theta}^Y,A)_{\eta_{(A,M(v)(Y))}}\c F((u,\id{}),A)\left(\eta_{(\id{A},M(\theta)_Y)}\right)$}$$
By the extraordinary naturality of $\eta$,
$$\eta_{(\id{A},M(\rho\c \theta)_Y)}=F((\id{A},M(\theta)_Y),A)\left(\eta_{(\id{A},M(\rho)_Y)}\right)\c \eta_{(\id{A},M(\theta)_Y)}$$
And by the uniqueness of the liftings of $2$-cells through $\groth{M}$, we have that
$$\underline{(\rho\c \theta)}^Y=\underline{\rho}^Y\c (\id{A},M(\rho)_Y)\h \underline{\theta}^Y.$$
So, by $2$-functoriality of $F$, it suffices to prove that
$$F(\underline{\theta}^Y,A)_{F((\id{A},M(\rho)_Y),A)\left(\eta_{(A,M(w)_Y)}\right)}\c F((u,M(\theta)_Y),A)\left(\eta_{(\id{A},M(\rho)_Y)}\right)$$
is equal to
$$F((v,\id{}),A)\left(\eta_{(\id{A},M(\rho)_Y)}\right)\c F(\underline{\theta}^Y,A)_{\eta_{(A,M(v)(Y))}}.$$
But this is true by naturality of $F(\underline{\theta}^Y,A)$ applied to the morphism
$$\eta_{(\id{A},M(\rho)_Y)}\:\eta_{(A,M(v)(Y))}\to F((\id{A},M(\rho)_Y),A)\left(\eta_{(A,M(w)(Y))}\right).$$
The fact that ${\left(\alpha_{(B,Y),f}\right)}_u$ is a natural transformation is checked with techniques similar to the above ones, noticing that
Whereas showing that ${\left(\alpha_{(g,\gamma),A}\right)}_u$ is a natural transformation uses that for $(g,\gamma)\:(B,Y)\to (B',Y')$
$$\underline{(\theta\h g)}^{Y}=\underline{\theta}^{M(g)(Y)}\h (g,\id{})\quad \text{and}\quad \underline{\theta}^{Y'}\h(\id{},\gamma)=(\id{},M(v)(\gamma))\h[1.5]\underline{\theta}^{M(g)(Y)}.$$
At this point, it is straightforward to check that $\alpha_{(B,Y),A}$ is $\oplaxn\mbox{-}\lax$ in $((B,Y),A)$. And it is immediatly seen that we obtain a bijection between the $\alpha$'s and the $\eta$'s by construction.

Finally, we extend such bijection to an isomorphism of categories. Given a modification $$\Theta_{(B,Y),A}\:\alpha_{(B,Y),A}\aR{}\beta_{(B,Y),A}\:\HomC{\A}{K(B,Y)}{A}\to F((B,Y),A)$$
between $\oplaxn\mbox{-}\lax$ natural transformations in $((B,Y),A)$, we send it to the modification
\begin{cd}[6][7]
1 \arrow[r,"{\id{B}}"]\&[-1ex]\HomC{\A}{B}{B}\arrow[r,bend left=27,"{\alpha_{(B,Y),B}}",""'{name=A}]\arrow[r,bend right=27,"{\beta_{(B,Y),B}}"',""{name=B}]\&F((B,Y),B)
\arrow[Rightarrow,from=A,to=B,shift right=0.5ex,"{\Theta_{(B,Y),B}}"{description,pos=0.46},shorten <=0.15ex,shorten >=0ex]
\end{cd}
between extraordinary lax natural transformations (that the latter is such is easily checked). And such assignment is surely functorial.

Given a modification
$$\Gamma_{(B,Y)}\:\eta_{(B,Y)}\aR{}{\eta'}_{(B,Y)}\:1\to F((B,Y),B)$$
between extraordinary lax natural transformations, we construct a corresponding modification $\Theta$. We see that, if we want to reach an isomorphism of categories, we need to define
$${\left(\Theta_{(B,Y),B}\right)}_{\id{B}}\deq \Gamma_{(B,Y)}.$$
Whence, given an arbitrary $u\:B\to A$ in $\A$, since we want $\Theta_{-,A}$ to be a modification, considering $(u,\id{})\:(B,Y)\to (A,M(u)(Y))$ in $\Grothop{M}$, we need to define
$${\left(\Theta_{(B,Y),A}\right)}_{u}\deq F((u,\id{}),A)\left(\Gamma_{(A,M(u)(Y))}\right).$$
It is straightforward to check that $\Theta$ is then a modification between $\oplaxn\mbox{-}\lax$ natural transformations. And such assignment is surely functorial. At this point, it is immediate to see that the two functors are, by construction, inverses of each other, giving the desidered isomorphism of categories.
\end{proof}

\begin{rem}
We are now ready to show that a pointwise left Kan extension in $\twoCATlax$ along a $2$-$\Set$-opfibration is always a weak left Kan extension.
\end{rem}

\begin{prop}\label{pointkanisalsoweak}
Consider a diagram
\begin{cd}[7][5]
\B \arrow[rr,"{F}"]\arrow[d,"{K}"']\&\arrow[dl,Rightarrow,shift right=0.75ex,"{\lambda}"{pos=0.355, inner sep= 0.35ex},shorten <=1.65ex,shorten >= 3.35ex]\& \C \\
\A\arrow[rru,"L"']
\end{cd}
in $\twoCATlax$ with $\B$ small and $K$ a $2$-$\Set$-opfibration. Assume that $\lambda$ exhibits $L=\Lan{K}{F}$ \pteor{in the sense of \defx\ref{defpointkanextensionlaxthreecat}}. Then $\lambda$ also exhibits $L=\lan{K}{F}$.
\end{prop}
\begin{proof}
Since $L=\Lan{K}{F}$, for every $C$, the $2$-cell
\begin{equation*}
\HomC{\A}{K(-)}{A}\aRR{L}\HomC{\C}{(L\c K)(-)}{L(A)}\aRR[\oplaxn]{\HomC{\C}{\lambda_{-}}{\id{}}}\HomC{\C}{F(-)}{L(A)};
\end{equation*}
is $2$-universal, giving an isomorphism of categories
\begin{equation}\label{eqisomorphismpointkanextnelladimcheeweak}
\HomC{\C}{L(A)}{C}\arr{}\HomC{\moplaxn{\B\op}{\CAT}}{\HomC{\A}{K(-)}{A}}{\HomC{\C}{F(-)}{C}}
\end{equation}

We need to prove that, for every $U\in \mlax{\A}{\C}$, pasting with $\lambda$ gives an isomorphism of categories
\begin{equation*}
\HomC{\mlax{\A}{\C}}{L}{U}\iso \HomC{\mlax{\B}{\C}}{F}{U\c K}.
\end{equation*}
So consider a lax natural transformation $\phi\:L\alax{} U$. For every $A\in \A$, the component $\phi_A\:L(A)\to U(A)$ corresponds to an oplax normal natural transformation
via the isomorphism of equation~\refs{eqisomorphismpointkanextnelladimcheeweak}. And $\phi_A$ being lax natural in $A\in \A$ precisely corresponds to the oplax normal natural transformations $\alpha_{-,A}$ being lax natural in $\A$, with structure $2$-cell on $f\:A\to A'$ in $\A$ given by the image of $\phi_f$ through the isomorphism of equation~\refs{eqisomorphismpointkanextnelladimcheeweak}. This means that the lax natural transformations $\phi$ precisely correspond to functors $\alpha_{B,A}$ $\oplaxn\mbox{-}\lax$ natural in $(B,A)\in \B\op\times \A$.

\noindent Consider then a modification $\Sigma\:\phi\aM{}\psi\:L\alax{}U$. The components $\Sigma_A$ with $A\in \A$ correspond to modifications $\Theta_{-,A}$ between oplax normal natural transformations $\alpha_{-,A}$ and $\beta_{-,A}$. And the modification axiom for $\Sigma$ corresponds to the modification axiom for $\Theta_{B,-}$ for every fixed $B\in \B$. So the modifications $\Sigma$ precisely correspond to modifications $\Theta$ between $\oplaxn\mbox{-}\lax$ natural transformations $\alpha$ and $\beta$. By the functoriality of the isomorphism of equation~\refs{eqisomorphismpointkanextnelladimcheeweak}, we obtain an isomorphism of categories between $\HomC{\mlax{\A}{\C}}{L}{U}$ and the category of $\oplaxn\mbox{-}\lax$ natural transformations
in $(B,A)\in \B\op\times\A$ and modifications between them.

By \thex\ref{parametrizedyonedaoplaxnlax} (the $\oplaxn\mbox{-}\lax$ parametrized Yoneda lemma), the latter category is then isomorphic to the category of extraordinary lax natural transformations
$$1\to \HomC{\C}{F(B)}{U(K(B))}$$
in $B\in \B$ and modifications between them, which is isomorphic (for example, by Hirata's paper~[8]) to $\HomC{\mlax{\B}{\C}}{F}{U\c K}$. Therefore we have produced an isomorphism of categories
\begin{equation*}
\HomC{\mlax{\A}{\C}}{L}{U}\iso \HomC{\mlax{\B}{\C}}{F}{U\c K},
\end{equation*}
and we can read that this is given by pasting with $\lambda$.
\end{proof}

\bibliographystyle{abbrv}
\begin{thebibliography}{10}

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\end{document}


# AI4GCC - Team: Below Sea Level
Critiques and Improvements

Bram Renting*
Leiden University
Delft University of Technology Phillip Wozny*
Tilburg University
Vrije Universiteit Amsterdam Robert Loftin
Delft University of Technology Claudia Wieners
Utrecht University Erman Acar
University of Amsterdam

###### Abstract

We present a critical analysis of the simulation framework RICE-N, an
integrated assessment model (IAM) for evaluating the impacts of climate change
on the economy. We identify key issues with RICE-N, including action masking
and irrelevant actions, and suggest improvements such as utilizing tariff
revenue and penalizing overproduction. We also critically engage with features
of IAMs in general, namely overly optimistic damage functions and unrealistic
abatement cost functions. Our findings contribute to the ongoing efforts to
further develop the RICE-N framework in an effort to improve the simulation,
making it more useful as an inspiration for policymakers.

##  1 Introduction

Over the course of implementing the first two submissions, we engaged with the
simulation framework RICE-N. We interrogate both RICE-N as implemented by
Zhang et al. [1] and the assumptions inherent to the class of models to which
it belongs, known as integrated assessment models (IAMs).

##### Executive summary

Our key points of criticism can be categorically summarised as follows:

  * •

Action masking inflates model performance

  * •

Issues with RICE-N:

    * –

Most actions are irrelevant for key climate and economic indices.

    * –

Tariffs do not impact reward of the affected state.

    * –

Trade does not impact reward of the affected state as intended.

    * –

Damages have little impact on reward.

  * •

Suggestions for RICE-N improvement:

    * –

Use tariff revenue.

    * –

Penalize overproduction.

    * –

Allow for technology sharing and wealth redistribution.

    * –

Let abatement cost depend on the previous mitigation level.

    * –

Strengthen the damage function.

  * •

Issues with IAMs:

    * –

The damage function is overly optimistic.

    * –

Abatement costs do not depend on previous mitigation.

##  2 Problems with Action Masking

Action masking is used to disable actions that are inconsistent with
negotiated agreements. Excessive masking can force protocols into seemingly
desirable behavior despite being the result of random behavior. This
phenomenon is evident in the Bilateral Negotiator released with the
competition. The protocol performs better after training for one episode than
after extensive training. This is due to the following well-intended design
features. First, agents commit to the maximum accepted proposal. With 27
agents sending and receiving proposals, one of the 54 total proposals likely
corresponds to a high mitigation rate. Second, masks are used to enforce the
maximum accepted proposal. As such, high levels of mitigation are almost
certainly enforced.

Finally, we argue that actions rendered inaccessible through masking are
unrealistic as states can never be forced to commit to their agreements.
Instead, collectively unfavorable actions should be made less desirable
through extrinsic factors imposed by other states.

##  3 Problems with the RICE-N model

###  3.1 Influence of actions on key measures

In order to explore the space of attainable outcomes in RICE-N we now analyze
the correlation of key performance measures to a range of possible action
inputs. In this context, we restrict the actions such that they are fixed for
a full simulation and that all regions perform the same action. We consider
RICE-N with 10 discrete action levels. There are 5 different types of actions
of which we want to sample every possible combination, resulting in a total of
$10^{5}$ environment rollouts. We collected the climate index, economic index,
and reward of the episode and generated the correlation matrix shown in Figure
1.

Figure 1: Correlation between RICE-N actions and performance measures. As can
be seen, climate and economic indexes only correlate with mitigation and
savings rates. Reward is more connected to actions.

We observe that there is practically no correlation between desired import,
tariff rate, maximum export, and climate/economic index. Only the mitigation
rate and savings rate have an impact on the economic index and the climate
index. Realize that this makes all the other actions irrelevant for the first
submission track of this competition. We further explore this by plotting the
results of every episode in Figure 2 and observe that the $10^{5}$ episodes
form $100$ perfectly overlapping dots. The difference between dots along the
$x$-axis represents the savings rate and along the $y$-axis the mitigation
rate. We also observe that optimizing for reward without any restriction
results in a low score both in the economic and climate index. Finally, we
would like to note that gross output, and thus the economic index, is an
internal affair that the states will not punish each other over. It is
unrealistic to expect states to force themselves to increase their gross
output as this would only lower their reward. Such issues make high-economic
index solutions not learnable.

This is not the case for climate mitigation as states do have the incentive to
punish other states that do not mitigate. There are two methods to punish non-
compliant states: (1) imposing tariffs on those states, and (2) limiting
imports from that state. However, Figure 1 suggests that there is barely an
effect of tariffs on the reward. Even more surprising is the negative
correlation between desired imports and the reward, which suggests that
limiting imports from a state is rewarding that state. We study this in more
depth in the following sections.

Figure 2: Results of manually sampling the RICE-N model with fixed actions.
All regions perform identical actions for all timesteps. Every possible
combination of actions is attempted and represented by a dot in this figure.
The dots perfectly overlap in 100 points changing position only based on
mitigation and savings rate. Of all the overlapping dots, the maximum reward
of that point is visualized by a color gradient.

####  3.1.1 Effects of tariffs on reward

The first sanctioning mechanism of the RICE-N model is the tariff. This is
informed by literature on climate economics, which uses tariffs and levies to
adjust the cost of carbon [2, 3]. In previous IAMs, there is a static
parameter corresponding to the loss of welfare per unit of tariff [4]. At each
step, agents compare the cost of mitigation to the expected loss of welfare
due to tariffs. Once the latter exceeds the former, mitigation becomes the
preferred action.

Currently, there is no global authority that can externally sanction
defection. As such, it is the responsibility of states to sanction one
another. The organizers’ implementation of the RICE-N model acknowledges this
fact and grounds the sanction in the trade dynamics of the simulation itself
[1]. As a result of that, there is no “loss of welfare due to tariff”
parameter. It is assumed that if agents apply tariffs to each other it will
result in a loss of revenue. However, as we will show empirically in the
following analysis, tariffs have negligible to no impact on reward in most
cases.

Figure 3: Comparison of the average normalized reward in each experimental
condition: pariah, control, and free trade. The pariah agent receives a fixed
tariff, the control agent follows the protocol, and the free trade agent has
no tariffs. We examine three different fixed tariff values, 5, 7, and 9. The
black diamonds represent outliers. Figure 4:  Visualization of the average
tariff rate of the agent in question confirms that our experimental conditions
work as intended.

##### Methods

To measure the effects of tariffs on reward and trade dynamics, we conducted
the following experiment comparing three agent groups: pariahs, controls, and
free trade. Pariah agents received a fixed tariff from all other agents,
control agents simply followed their trained protocol, and free trade agents
received no tariff. We examined three different fixed tariff values for the
pariah agents; 5, 7, and 9. Using a model trained without negotiation for
$100\,000$ episodes, we ran the simulation 1000 times for each group and each
experimental condition.

During each iteration of the experiment, an agent was chosen uniformly at
random as the subject. Between each group and each fixed tariff value, the
agent and environment were reset. As such, each experimental condition
contains data from all agents. As the reward varies considerably between
states and is relatively stable within states, we normalize the reward by each
state id. Therefore, we compare the effect of tariffs on rewards relative to
each state.

In a follow-up experiment, we sampled the RICE-N model with fixed actions that
represent an ideal trade scenario where the mitigation rate is $0.9$, savings
rate $0.3$, desired import $0.9$, and maximum export $0.9$. Under such high
trade conditions, the effect of tariffs would be expected to be magnified. We
compared the rewards of each state with both maximum tariffs and no tariffs at
all in order to measure the effect of tariffs on reward.

##### Results

As visible in Figure 3, manipulating tariff magnitude does not result in
significant differences in relative reward. To ensure that our tariff
manipulation is effective, we also gathered the average tariff per subject in
each treatment condition (see Figure 4). Taken together, it indicates neither
the maximum tariff nor the absence of tariff from all agents towards any
single agent impacts that agent’s reward. This suggests a flaw in the trade
component of the reward calculation.

The second experiment showed that the reward of a state is untouched by other
states’ tariff actions. Even more surprising is that the reward of the state
that imposed the tariff decreases by $0.02-6.50\%$.

##### Explanation

For the RICE-N model to accurately model interstate commerce and climate-
related tariff mechanisms, the tariff mechanism needs to impact reward. The
failure of tariffs to impact the reward can be traced back to how trade
influences the reward which we explain in the following.

The reward is calculated as aggregate consumption which can be decomposed into
two additive terms, foreign and domestic consumption, and trade is the amount
of import/export from one country to another. Critically, trade takes two
forms, scaled imports, and tariffed imports. The former is the overlap between
the desired imports of one state corrected for gross output and the exporting
capacity of the other. The latter adjusts the scaled imports by the inverse of
the tariff. Foreign consumption utilizes tariffed imports and domestic
consumption utilizes scaled imports. Therefore, if state A tariffs state B, it
is only factored into the reduction of state A’s foreign consumption term.
State B’s domestic consumption, which depends on its export to State A,
remains unchanged by the tariff. Therefore, tariffing a state does not affect
the reward of the state being tariffed.

####  3.1.2 Effects of desired imports on reward

We now focus on the desired imports and maximum exports actions, or simply the
amount of trade. We sampled the RICE-N model with fixed actions where the
mitigation rate is $0.9$, savings rate $0.3$, and tariffs $0$ to maximize the
benefit of trade. We varied the desired imports and maximum export actions,
observing the reward.

Reward increases for all states when trade is limited. If we compare the
extreme cases of maximum trade and no trade, the reward per state increases by
a percentage of $1-2322\%$, depending on the state, when no trade is
happening. It appears that limiting imports cannot be used to negatively
impact the reward of another state, but actually positively impacts it. This
is likely the case because domestic consumption is preferred over foreign
consumption in the current setup of RICE-N. As export lowers your domestic
consumption and the import does not compensate for this, states will prefer
not to export.

####  3.1.3 State punishment potential

We can conclude that neither imposing tariffs nor limiting imports are
suitable sanction mechanisms in RICE-N. Moreover, limiting imports even
increases the reward of the subjected state. As such, states have no leverage
with which to negotiate. Any observed form of “learned” negotiation is likely
a result of random behavior or favorable masking or both. Any optimal learning
agent will end up at the point of maximum reward (see Figure 2) at a climate
index of 0.33 and an economic index of 0.6. Therefore, the only mechanism to
obtain more optimal policies is to exploit the RICE-N framework itself.

###  3.2 Our submission one

Due to these shortcomings, we _strategically_ utilized three Bilateral
Negotiator variations which make use of the action masking quirk described in
Section 2. That is, the protocols were essentially untrained. One protocol
made use of favorable masking for savings, another for mitigation, and the
third for both savings and mitigation. The resulting Pareto frontier is
visible in Figure 5. We are aware of the fact that such protocols are not in
the spirit of the competition. However, it was stated during the first
workshop session that tracks one and two submissions may be independent and
that the goal of track one is purely metric optimization.

Figure 5: We compare three variations on the Bilateral Negotiator protocol.
The name of each protocol indicates the action which is bilaterally
negotiated. Each of these protocols was trained for one episode. Note, these
three submissions are the same used for Track 1.

###  3.3 Finite horizon

When no mitigation is performed by states, the damages reduce the gross output
of states by $\sim 8.5\%$. This is only a minor deduction of the gross output
and does not reflect the real-world _sword of Damocles_ which is a climate
disaster. From the perspective of the agents, the game simply ends and no
large damages are achieved. Reducing an infinite horizon game to a finite one
changes optimal strategies from a game-theoretical perspective.

We extended the horizon of the simulation to 200 and 300 years to let damages
influence the agents more heavily. We trained agents without a protocol with
the hope that major future damages would cause the agents to learn to
mitigate. However, while damages rose to $\sim 13\%$ and $\sim 22\%$,
mitigation rates remained as before. We also saw earlier in Figure 1 and
Figure 2 that reward negatively correlates with gross output (economic index).
We can conclude that there is little effect of temperature rise on state
reward.

##  4 Suggestions for improvement

###  4.1 Tariff revenue

Currently, tariff revenue is calculated during the climate economic simulation
step but not used. Tariff revenue could contribute to government balance, the
debt ratio, and the subsequent import capacity for the next climate economic
simulation step. Literature on climate policy-induced economic inequality
suggests that carbon tariff revenue should be reinvested in developing
countries to develop climate mitigation infrastructure [5].

###  4.2 Cost of overproduction

As stated in Section 3.1.1, tariffs do not impact reward. A possible
workaround could make use of scaled and tariffed imports. The differences
between the two correspond to the amount of output produced for export but not
ultimately consumed by the importer. Currently, the exporting country faces no
penalty for overproduction. Adding the difference between tariffed imports and
scaled imports to the export term of domestic consumption would lower overall
consumption and function as a penalty to the country being tariffed.

###  4.3 Disaster influenced reward

Future damages should have an impact on the reward of states, which seems to
be limited currently. The presence of a horizon in the simulation creates a
situation without a doomsday scenario. Simulating an infinite game is of
course impossible, but adding an artificial _sword of Damocles_ might help.
This could be in the form of a high negative reward when a certain temperature
rise is passed.

##  5 Issues with IAMs

###  5.1 Damage by climate change

DICE’s damage function describes the fraction of GDP lost to climate damage.
It is unlikely that catastrophic climate conditions of a 5°C temperature rise
would only result in a 5% loss in GDP [6]. As such, more realistic damage
functions are required, such as Weitzman [7] which allows for an “infinitely
bad” climate, or Burke et al. [8], which attempts to construct a damage
function from empirical data.

Damage functions can also be viewed as secondary to the “guardrail approach”,
which emphasizes a target warming threshold. Policymakers then focus on
staying under the threshold as opposed to avoiding damages [9]

###  5.2 The dynamics of mitigation costs

Currently, mitigation costs persistently depend only on the current mitigation
level. In reality, mitigation costs are transitional. Constructing a wind farm
is costly during construction, but not so once completed. Persistent
mitigation costs incentivize the latter investment, once abatement costs are
decreased due to technology. In contrast, transitional mitigation costs
incentivize early investment, as that reduces the cost of subsequent
mitigation.

Grubb et al. [10] proposed a correction to DICE which includes a transitional
mitigation cost function. This allows states to negotiate around rates of
change as opposed to absolute mitigation levels which is more in line with
real-world climate targets (e.g. halving emissions by 2030).

## Acknowledgments and Disclosure of Funding

We would like to thank Maikel van der Knaap, Cale Davis, Albert Bomer,
Catholijn Jonker, and Holger Hoos for their time spent discussing various
topics of this competition.

This research was (partly) funded by the Hybrid Intelligence Center, a 10-year
programme funded by the Dutch Ministry of Education, Culture and Science
through the Netherlands Organisation for Scientific Research, grant number
024.004.022.

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