Similar questions have already been asked [here](https://mathoverflow.net/q/268743/18238) and [here](https://mathoverflow.net/q/53548/18238) but not exactly in the direction I need.


I have a (small) index category $\mathcal{I}$ which is **not** cofiltered, and I need to consider categories of projective systems, indexed over $\mathcal{I}$, with values in some $R-\mathbf{Mod}$ (for several rings $R$, all commutative). I won't need to change $\mathcal{I}$, so I don't need a whole theory about all possible weird projective systems, but I am trying to find references for things like


* The categories of $R-\textbf{Mod}$-valued $\mathcal{I}$-projective systems is abelian;
* The componentwise tensor-product is a direct limit;
* The componentwise tensor-product satisfies the natural adjunction with respect to an internal $\operatorname{Hom}$;
* A description of injective/projective objects (in particular, of flat objects to study commutativity of taking componentwise tensor-product and inverse limit over $\mathcal{I}$).


All references I know, SGA 4 *in primis*, require that $\mathcal{I}$ is filtered, but this is certainly not my case. I would like to avoid reinventing the wheel (expecially because I can end up with a square one), and would be very grateful if someone knows references for the above facts or knows if they are true/false beyond the filtered case.



Need of filtered indexed categories
The fact that categories of projective (or inductive) systems over arbitrary categories in abelian categories are again abelian is explained in Grothendieck's Tohoku paper (Sections 1.6 and 1.7).


