This is quite a broad question regarding constructions of categories of sheaves in geometry.
Let $\textbf{Sch}$ denote the category of schemes. Let $\textbf{SchAff}$ denote the full subcategory of affine schemes. Recall that this category is canonically isomorphic to the opposite of the category of commutative rings, which we denote by $\textbf{ComR}$. To any $X\in \textbf{Sch}$, we have the Abelian category of quasi-coherent $\mathcal{O}_X$-modules, which we denote by $\mathbf{QCoh(X)}.$ For an affine scheme $Spec(R)$ this is just the category of $R$-modules. Pullback allows us to arrange all this data into a 2-functor:
$$f^*: \textbf{Sch}^{op} \rightarrow \textbf{Ab},$$
where $\textbf{Ab}$ denotes the 2-category of Abelian categories. If we restrict this functor to $\textbf{ComR}$ it is particularly easy to understand: it sends a ring to its category of modules and pullback is just tensor product.
This type of construction occurs all the time in geometry. For example, instead of quasi-coherent sheaves, we could take etale, or $\ell$-adic sheaves. If we restricted to smooth, complex schemes over $Spec(\mathbb{C})$ we could take $D$-modules. If we restrict to schemes over a finite field we could take arithmetic $D$-modules.
My first question is the following:
After strictifying, are all of these pullback functors right Kan-extensions from their restriction to commutative rings? If we took pushforward instead of pullback would they be left Kan-extensions?
We could also beef things up to the derived world. For example, in the quasi-coherent case, pullback would give the $\infty$-functor:
$$f_{\infty}^*: N(\textbf{Sch}^{op}) \rightarrow \textbf{Stab}_{\infty},$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; X \rightarrow D(X),$$ where $\textbf{Stab}_{\infty}$ is the $\infty$-category of stable $\infty$-categories, $N(\textbf{Sch}^{op})$ is the nerve, and $D(X)$ is the ($\infty$-categorical) derived category of quasi-coherent sheaves on $X$.
As above, this basic construction can be carried out in many different settings.
My second question is the following:
Are all of these pullback $\infty$-functors right Kan-extensions (in the $\infty$-categorical sense) from their restriction to $N(\textbf{ComR})$?