I've been wondering about the following: Is there a $\textit{neat}$ description of the category $[\mathcal P]\mathcal{Mod}(\mathcal O)$ of [pre]sheaves of modules on a sheaf of rings $\mathcal O$?
Let me describe what I'd ideally want to see by using the example of sheaves of abelian groups.
Here the category of presheaves of abelian groups on a (let's restrict to topological spaces for now, the variant for sites shouldn't be too different I guess) space $X$ is nothing but the functor category $[\mathcal{Ouv}(X)^{op}, \mathcal{Ab}]$ where $\mathcal{Ouv}(X)$ is the [category associated to] the poset of open sets on $X$.
I think this is quite useful. It not only immediately tells us that the category of presheaves of abelian groups is $\textit{bicomplete}$ but also how to effectively compute limits/colimits in this category.
This is useful when combined with $\textit{sheafification}$ for example. We know that there is a forgetful functor $for: \mathcal{Sh}_X(\mathcal{Ab}) \to \mathcal{Psh}_X(\mathcal{Ab})$ and that sheafification $(\cdot)^{sh}$ is a left adjoint to this functor.
This tells us for example, that $(\cdot)^{sh}$ commutes with colimits. So that one deduces $\textit{formally}$ (only using that $(\cdot)^{sh} \circ for \cong id$) how colimits in the category of sheaves of abelian groups look like: They are given as sheafification of the respective colimit in the category of presheaves (which are once again easy to handle).
Moreover, $for$ commutes with limits telling us that one knows that a limit of sheaves looks like the presheaf limit $\textit{if it exists}$.
I know that these considerations basically stay true in the case of $\mathcal O$-modules, but I don't know if there is a way seeing this that is similarly simple as in the case of sheaves of abelian groups.
Is it possible to define $\mathcal{PMod}(\mathcal O)$ in a similar fashion as functors into some category where one has a fairly good understanding of how limits/colimits look like etc.?
I'd assume it'd be something like a category with objects $(R, M)$ where $R$ is a ring and $M$ a $R$-module and where a morphism $(R,M) \to (S,N)$ is given by a ring morphism $R \to S$ and a module homomorphism $M \to (R \to S)_{*}N$ or something like that.
I'm not sure however if this kind of construction is considered somewhere else more "abstractly" giving a neat description of its limits/colimits with $\textit{less}$ work than it would be for the ordinary definition of presheaves of modules.