I know the fundamental facts about rings and modules over rings and I understand how they form their respective categories. What I was wondering about is the relationship between $\operatorname{R-mod}$ and $\operatorname{Ring}$.
It is clear to me that any ring $R\in\operatorname{Ring}$ gives raise to its own category $\operatorname{R-mod}$. Anyhow, an equivalent definition of $R$-module is to consider some $M\in\operatorname{Ab}$ together with a morphism $\mu\in\operatorname{Ring}(A,End_{\mathbb{Z}}(M))$. Where $End_{\mathbb{Z}}(M)$ is the ring of group endomorphisms $M$ with operations of pointwise sum, composition product, and unity given by the identity morphism on $M$.
So I was wondering if we can realize all the categories $\{\operatorname{R-mod}\}_{R\in\operatorname{Ring}}$ with some kind of "slice under $R$" construction. It seems to me that the 'basic' one does not work here since we have to consider only some codomain objects and moreover there are problems with the notion of morphism.
If not is there a precise way we can formalize and abstract the construction of the categories $\{\operatorname{R-mod}\}_{R\in\operatorname{Ring}}$ from $\operatorname{Ring}$? Are there interesting facts about their relationship?
Any help or reference would be great!
There's a category with objects $(R,M)$, $R$ a ring and $M$ an $R$-module with morphisms $(R,M)\to(S,N)$ consisting of a ring homomorphism $\phi:R\to S$ and a group homomorphism $\psi:M\to N$ satisfying $$\psi(rm)=\phi(r)\psi(m).$$ There is an obvious functor to $\text{Ring}$ and the fibre at a ring $R$ is $R$-mod.