I have a ring $A$ (specifically a valuating ring) with a non-trivial ideal $I$. I have shown that we can identify $\text{Mod}_{A/I}$ with the strictly full additive subcategory $\mathcal{C}$ of $\text{Mod}_A$ consisting of those $A$-modules $M$ with $IM=0$.
Is there a quick and easy way to see that translating between taking tensors over $A/I$ and over $A$ works exactly as one would hope? In other words, that if $M$ and $N$ are objects in $\mathcal{C}$, then $M\otimes_A N=M\otimes_{A/I}N$ and the same for tensoring morphisms.