Condition for a functor to induce a cartesian closed functor between categories of presheaves

Question

We denote the category of presheaves on a small category ${\cal C}$ (set-valued functor-category) by $$\widehat{\cal C}:=[{\cal C}^{op},{\bf Set}].$$ Such a category is cartesian closed, i.e. it has a terminal object $T$, any two objects $X$ and $Y$ of $\widehat{\cal C}$ have a product $X\times Y$ in $\widehat{\cal C}$ and any two objects $Y$ and $Z$ have an exponential $Z^Y$ in $\widehat{\cal C}$ with natural bijections $$\widehat{\cal C}(X\times Y, Z)\cong \widehat{\cal C}(X,Z^Y).$$ Let $F\colon {\cal C}^{op} \to {\cal D}^{op}$ be any functor between small categories and $$F^\ast\colon [{\cal D}^{op},{\bf Set}] \to [{\cal C}^{op},{\bf Set} ]$$ the induced functor, given by precomposition with $F$.

Question: Under which exact conditions on $F$ is its induced functor $F^\ast$ cartesian closed, i.e. such that the natural comparison morphism $$\gamma_{X,Y}\colon F^{\ast}(Y^X)\to F^{\ast}(Y)^{F^\ast(X)}$$

is an isomorphism for all objects $X$ and $Y$ of $[{\cal D}^{op},{\bf Set}] $.

In more details: Thanks to the fact, that $F^{\ast}$ preserves products, since it has the left Kan-extension ${\rm Lan}_{F}\colon [{\cal C}^{op},{\bf Set} ] \to [{\cal D}^{op},{\bf Set}] $ as a left adjoint, we can form the morphism $$F^\ast(Y^X)\times F^\ast(X) \cong F^{\ast}(Y^X\times X)\stackrel{F^{\ast}({\rm ev})}{\longrightarrow} F^{\ast}(Y),$$ which has the morphism $$\gamma_{X,Y}\colon F^{\ast}(Y^X)\to F^{\ast}(Y)^{F^\ast(X)}$$ in question as an adjoint.

Answer 1

Here is an observation you already almost made: since $F^*$ has a left adjoint $F_!$ given by left Kan extension, $F^*$ preserves exponential objects if and only if $F_!$ preserves products.

Note that $F_!$ is given by an unwieldy colimit so this does not always happen. Here is however a natural condition which should make it work: in $\mathrm{Set}$, finite products commute with sifted colimits (see https://ncatlab.org/nlab/show/sifted+colimit), so if you can guarantee that all the indexing categories of the colimit defining the left Kan extension are sifted, $F_!$ will preserve products.

In particular, categories with finite products are cosifted (and this plays nicely with the $\mathrm{op}$ of presheaves) so that if $F$ is a product-preserving functor between categories with finite products, $F^*$ preserves exponential objects.