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Perhaps something like right Kan extension along the inclusion of the (bi)category of presheaf toposes into the (bi)category of Grothendieck toposes might work – but the existence of toposes with no points suggests it may not – but it would be nice to have a somewhat more concrete description.

Since I am looking for a construction where $\mathcal{A}$ is not necessarily the category of models of a geometric theory, I conclude that I cannot require the category of presheaves on $\mathcal{C}$ to be $[\mathcal{C}^\textrm{op}, \mathcal{A}]$.

In any case, pointwise right Kan extension along the inclusion of the (bi)category of presheaf toposes into the (bi)category of Grothendieck toposes does not work: given $\mathcal{A} (\textbf{Psh} (\mathcal{C}))$ for all presheaf toposes $\textbf{Psh} (\mathcal{C})$, the extension is simply $$\tilde{\mathcal{A}} (\mathcal{E}) = \int_{\textbf{Psh} (\mathcal{C})} [\textbf{Topos} (\textbf{Psh} (\mathcal{C}), \mathcal{E}), \mathcal{A} (\textbf{Psh} (\mathcal{C}))]$$ but if $\mathcal{E}$ is a topos with no points, then there are no geometric morphisms $\textbf{Psh} (\mathcal{C}) \to \mathcal{E}$ either (unless $\mathcal{C} = \emptyset$). Even for a Hausdorff space $X$, we find that $\tilde{A} (\textbf{Sh} (X)) \simeq \mathcal{A} (\textbf{Set}^X)$, because geometric morphisms $\textbf{Psh} (\mathcal{C}) \to \textbf{Sh} (X)$ factor through $\textbf{Set}^X$ uniquely up to unique isomorphism. (In other words, presheaf toposes cannot "see" the topology of a Hausdorff space. This is perhaps easier to understand in the localic case, because localic presheaf toposes correspond to preordered sets equipped with their Alexandrov topology.)

There is a temptation to strengthen desideratum 6 to require that the category of presheaves on a (small) category $\mathcal{C}$ be equivalent to the category of functors $\mathcal{C}^\textrm{op} \to \mathcal{A}$, but this does not appear to be a good idea. As Simon Henry remarks, if $\mathcal{C}^\textrm{op}$ is filtered, then the unique functor $\mathcal{C} \to \mathbf{1}$ is a morphism of sites corresponding to a geometric morphism that has a right adjoint (i.e. the inverse image functor itself has a left adjoint that preserves finite limits), so contravariant (bi)functoriality with respect to geometric morphisms forces the induced $\mathcal{A} \to [\mathcal{C}^\textrm{op}, \mathcal{A}]$ to have a left adjoint, i.e. $\mathcal{A}$ must have colimits of shape $\mathcal{C}$. This is the same argument that shows that the category of points of a topos must have filtered colimits.

There is a temptation to strengthen desideratum 6 to require that the category of presheaves on a (small) category $\mathcal{C}$ be equivalent to the category of functors $\mathcal{C}^\textrm{op} \to \mathcal{A}$, but this does not appear to be a good idea. As Simon Henry remarks, if $\mathcal{C}^\textrm{op}$ is filtered, then the unique functor $\mathcal{C} \to \mathbf{1}$ is a morphism of sites corresponding to a geometric morphism that has a right adjoint (i.e. the inverse image functor itself has a left adjoint that preserves finite limits), so contravariant (bi)functoriality with respect to geometric morphisms forces the induced $\mathcal{A} \to [\mathcal{C}^\textrm{op}, \mathcal{A}]$ to have a left adjoint, i.e. $\mathcal{A}$ must have colimits of shape $\mathcal{C}^\textrm{op}$. This is the same argument that shows that the category of points of a topos must have filtered colimits.

1. The properties of limits and colimits in the category of sheaves on a general site with values in $\mathcal{A}$ are "similar" to those of $\mathcal{A}$ itself. (I am being vague here because even when $\mathcal{A}$ is locally finitely presentable, the category of sheaves with values in $\mathcal{A}$ may not be locally finitely presentable – this already happens for $\mathcal{A} = \textbf{Set}$.)
2. The category of sheaves on a site $(\mathcal{C}, J)$ with values in $\mathcal{A}$ is (pseudo)functorial in $(\mathcal{C}, J)$ with respect to morphisms of sites.
3. The construction respects Morita equivalence of sites, i.e. factors through the (bi)category of Grothendieck toposes.
4. The construction respects "good" (bi)colimits in the (bi)category of Grothendieck toposes, i.e. sends them to (bi)limits of categories. (I don't know what "good" should mean here, but at minimum it should include coproducts. When $\mathcal{A}$ is locally finitely presentable, there is a classifying topos, so in fact the construction respects all (bi)colimits.)
5. The category of sheaves on the point with values in $\mathcal{A}$ is canonically equivalent to $\mathcal{A}$.
6. The category of sheaves on the Sierpiński space with values in $\mathcal{A}$ is canonically equivalent to the arrow category of $\mathcal{A}$.

(If I'm not mistaken, assuming a sufficiently strong form of desideratum 4, desiderata 5 and 6 force the category of presheaves on $\mathcal{C}$ with values in $\mathcal{A}$ to be equivalent to the category of functors $\mathcal{C}^\textrm{op} \to \mathcal{A}$. I am not sure whether this should be an explicit desideratum.)

• ... when $\mathcal{A}$ is finitely accessible, not necessarily cocomplete, e.g. the category of Kan complexes?
• ... when $\mathcal{A}$ is an abelian category, not necessarily accessible, e.g. the category of finite abelian groups?
• ... when $\mathcal{A}$ is a Grothendieck abelian category, not necessarily locally finitely presentable?

Perhaps something like right Kan extension along the inclusion of the (bi)category of presheaf toposes into the (bi)category of Grothendieck toposes might work – but the existence of toposes with no points suggests it may not – but it would be nice to have a somewhat more concrete description.

1. The properties of limits and colimits in the category of sheaves on a general site with values in $\mathcal{A}$ are "similar" to those of $\mathcal{A}$ itself. (I am being vague here because even when $\mathcal{A}$ is locally finitely presentable, the category of sheaves with values in $\mathcal{A}$ may not be locally finitely presentable – this already happens for $\mathcal{A} = \textbf{Set}$.)
2. The category of sheaves on a site $(\mathcal{C}, J)$ with values in $\mathcal{A}$ is (pseudo)functorial in $(\mathcal{C}, J)$ with respect to morphisms of sites. (By "morphism of sites" I mean the notion that contravariantly induces geometric morphisms.)
3. The construction respects Morita equivalence of sites, i.e. factors through the (bi)category of Grothendieck toposes.
4. The construction respects "good" (bi)colimits in the (bi)category of Grothendieck toposes, i.e. sends them to (bi)limits of categories. (I don't know what "good" should mean here, but at minimum it should include coproducts. When $\mathcal{A}$ is locally finitely presentable, there is a classifying topos, so in fact the construction respects all (bi)colimits.)
5. The category of sheaves on the point with values in $\mathcal{A}$ is canonically equivalent to $\mathcal{A}$.
6. The category of sheaves on the Sierpiński space with values in $\mathcal{A}$ is canonically equivalent to the arrow category of $\mathcal{A}$.

• ... when $\mathcal{A}$ is finitely accessible, not necessarily cocomplete, e.g. the category of Kan complexes, or the category of divisible abelian groups?
• ... when $\mathcal{A}$ is an abelian category, not necessarily accessible, e.g. the category of finite abelian groups, or the category of finitely generated abelian groups?
• ... when $\mathcal{A}$ is a Grothendieck abelian category, not necessarily locally finitely presentable?

Perhaps something like right Kan extension along the inclusion of the (bi)category of presheaf toposes into the (bi)category of Grothendieck toposes might work – but the existence of toposes with no points suggests it may not – but it would be nice to have a somewhat more concrete description.

There is a temptation to strengthen desideratum 6 to require that the category of presheaves on a (small) category $\mathcal{C}$ be equivalent to the category of functors $\mathcal{C}^\textrm{op} \to \mathcal{A}$, but this does not appear to be a good idea. As Simon Henry remarks, if $\mathcal{C}^\textrm{op}$ is filtered, then the unique functor $\mathcal{C} \to \mathbf{1}$ is a morphism of sites corresponding to a geometric morphism that has a right adjoint (i.e. the inverse image functor itself has a left adjoint that preserves finite limits), so contravariant (bi)functoriality with respect to geometric morphisms forces the induced $\mathcal{A} \to [\mathcal{C}^\textrm{op}, \mathcal{A}]$ to have a left adjoint, i.e. $\mathcal{A}$ must have colimits of shape $\mathcal{C}$. This is the same argument that shows that the category of points of a topos must have filtered colimits.

Since I am looking for a construction where $\mathcal{A}$ is not necessarily the category of models of a geometric theory, I conclude that I cannot require the category of presheaves on $\mathcal{C}$ to be $[\mathcal{C}^\textrm{op}, \mathcal{A}]$.