Let $T$ be a small category, and $\mathrm{Mod}(T)\subset\mathrm{Fun}(T,\textbf{Set})$ the category of cartesian (finite-limit preserving) copresheaves on $T$. If $T$ is a commutative algebraic theory, then $\mathrm{Mod}(T)$ is cartesian closed (see commutative algebraic theory in nLab).

First question: Is the converse implication true? I mean: if $\mathrm{Mod}(T)$ is cartesian closed, can we prove that $T$ commutative?

Furthermore, if $T$ is the full subcategory of the simplicial category $\Delta$ with objects $[0], [1], [2]$ (where $[n]$ is the order $0<1<\dotsm<n$) we have that $\mathrm{Mod}(T)= \textbf{Cat}$ is cartesian closed, in fact, such a $T$ is representable as a monoid in the cartesian-monoidal category $\mathrm{Cat}^{\mathrm{op}}\downarrow ([0]\times [0])$ (equivalently, objects are spans to $[0]$ and $[0]$ in $T$, the monoidal product is by pullbacks) and the image of a model $M$ is just a monoid in $\textbf{Set}\downarrow C_0\times C_0$ (where $C_0=M([0])$), then a small category with $C_0$ as class object. Analogous argument for functors.

Second question: Does there exist a law to recognize from the diagram structure of $T$ if $\mathrm{Mod}(T)$ is cartesian closed?

Let $T$ be a small category, and $\mathrm{Mod}(T)\subset\mathrm{Fun}(T,\textbf{Set})$ the category of cartesian (finite-limit preserving) copresheaves on $T$. If $T$ is a commutative algebraic theory, then $\mathrm{Mod}(T)$ is cartesian closed (see commutative algebraic theory in nLab).

First question: Is the converse implication true? I mean: if $\mathrm{Mod}(T)$ is cartesian closed, can we prove that $T$ commutative?

Furthermore, if $T$ is the full subcategory of the simplicial category $\Delta$ with objects $[0], [1], [2], [3]$ (where $[n]$ is the order $0<1<\dotsm<n$) we have that $\mathrm{Mod}(T)= \textbf{Cat}$ is cartesian closed, in fact, such a $T$ is representable as a monoid in the cartesian-monoidal category $\mathrm{Cat}^{\mathrm{op}}\downarrow ([0]\times [0])$ (equivalently, objects are spans to $[0]$ and $[0]$ in $T$, the monoidal product is by pullbacks) and the image of a model $M$ is just a monoid in $\textbf{Set}\downarrow C_0\times C_0$ (where $C_0=M([0])$), then a small category with $C_0$ as class object. Analogous argument for functors.

Second question: Does there exist a law to recognize from the diagram structure of $T$ if $\mathrm{Mod}(T)$ is cartesian closed?

Let  $T$ a small category, and $Mod(T)\subset Fun(T, Set)$ the category of cartesian (finite limit preserving) copresheaves on $T$. If $T$ is a commutative algebraic theory then $Mod(T)$ is cartesian closed (see commutative algebraic theory in nLab  ).

First question: Is true the opposite implication? I mean: if $Mod(T)$ is cartesian closed, can we prove that $T$ commutative?

Furthermore if $T$ is the full subcategory of simplicial category  $\Delta$ with element $[0], [1], [2]$ (where $[n]$ is the order $0<1,\ldots <n$) we have that $Mod(T)= Cat$ is cartesian closed, infact a such $T$ is representable as a monoid of the cartesian-monoidal category $Cat^{op}\downarrow ([0]\times [0])$ (equivalently objects are spans to $[0]$ and $[0]$ in $T$, the monoidal product is by pullbacks) and the image of a model $M$ is just a monoid in $Set\downarrow C_0\times C_0$ (where $C_0=M([0])$), then a small category with $C_0$ as class object  . Analogous argument for functors.

Second question: exist a law for recognize from the diagram structure of $T$, if $Mod(T)$ is cartesian closed?

Let $T$ be a small category, and $\mathrm{Mod}(T)\subset\mathrm{Fun}(T,\textbf{Set})$ the category of cartesian (finite-limit preserving) copresheaves on $T$. If $T$ is a commutative algebraic theory, then $\mathrm{Mod}(T)$ is cartesian closed (see commutative algebraic theory in nLab).

First question: Is the converse implication true? I mean: if $\mathrm{Mod}(T)$ is cartesian closed, can we prove that $T$ commutative?

Furthermore, if $T$ is the full subcategory of the simplicial category $\Delta$ with objects $[0], [1], [2]$ (where $[n]$ is the order $0<1<\dotsm<n$) we have that $\mathrm{Mod}(T)= \textbf{Cat}$ is cartesian closed, in fact, such a $T$ is representable as a monoid in the cartesian-monoidal category $\mathrm{Cat}^{\mathrm{op}}\downarrow ([0]\times [0])$ (equivalently, objects are spans to $[0]$ and $[0]$ in $T$, the monoidal product is by pullbacks) and the image of a model $M$ is just a monoid in $\textbf{Set}\downarrow C_0\times C_0$ (where $C_0=M([0])$), then a small category with $C_0$ as class object. Analogous argument for functors.

Second question: Does there exist a law to recognize from the diagram structure of $T$ if $\mathrm{Mod}(T)$ is cartesian closed?