Defining a quasi-coherent module $\mathcal{M}$ on a scheme $X$ to be a compatible family of modules $(\mathcal{M}(x))_{x \in X(A), A \in \textbf{Rings}}$ (as in here), is there a straightforward way to show the existence of (finite) limits (and that it forms an abelian category)?

One possible way, of course, should be to show that this definition gives rise to a category equivalent to the category of quasi-coherent sheaves of modules on the small Zariski-site associated to $X$, but that feels like a rather dirty solution.

The problem, I guess, is that taking pullbacks of sheaves of modules (generally) doesn't commute with taking limits so that the limit isn't defined "fibrewise"; colimits work fine for exactly that reason.

Another argument that a friend of mine explained to me seems to be that, denoting the in the above way defined category of modules as $\textbf{Mod}(X)$, one has

$$\textbf{Mod}(X) = \varprojlim_{A \in \textbf{Aff}/X} \textbf{Mod}(A)$$

where the ($2$-)limit is taken in the $(2,1)$-category of categories, functors and natural isomorphisms.

Now the argument would be that $\textbf{Mod}(A)$ is a locally presentable category, (certain?) limits of locally presentable categories are locally presentable, and locally presentable categories admit arbitrary limits.

I was still wondering whether there wouldn't be a more elementary way to for example directly construct kernels and finite products of modules when defined this way.

I'd appreciate any thoughts!

//Edit: Ok another way seems to be to first show that one can glue quasi-coherent modules along Zariski-coverings and then do everything locally. I guess that's fine for me, but I'd still be interested in seeing other elementary arguments if anyone has one!

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