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](https://ncatlab.org/nlab/show/quasicoherent+sheaf#AsSheavesII)), 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!



Existence of finite limits of quasi-coherent modules on a scheme
Here is the precise statement alluded to in the comments:



> 
> Let $C = \lim\_i C\_i$ be a limit of categories
> with projections $\pi\_i : C \to C\_i$. Let $\{X\_j\}\_j$ be a
> diagram in $C$. If for every $i$ the induced diagram
> $\{\pi\_i(X\_j)\}\_j$ in $C\_i$ has a limit $X\_i$, and the
> transition functors $C\_i \to C\_{i'}$ send $X\_i \mapsto X\_{i'}$ for every
> morphism $i \to i'$ in the indexing category, then the original diagram
> $\{X\_j\}\_j$ in $C$ admits a limit $X$ such that $\pi\_i(X) = X\_i$ for every $i$.
> 
> 
> 


For example, if $X$ is a scheme, then quasi-coherent $O\_X$-modules can be defined as you did, except that you can require the maps $Spec(A) \to X$ to be Zariski immersions (since $X$ is a scheme). Then the transition functors are pullbacks along open immersions which are exact hence preserve finite limits.


Alternatively by descent, you can take a Zariski cover of $X$ by affines $U\_i$, then $Mod(X)$ will be a limit of $Mod(U\_i)$ and of the intersections (similar to the usual sheaf condition except that you have to go to the 3-way intersections since it is a sheaf of categories). Again you can then apply the same argument to say that limits will be computed on the $U\_i$'s.


