I have a commutative unital ring $A$ and a full abelian subcategory $\mathcal{C}$ of $\text{Mod}_A$. I need to calculate $\text{Ext}(M,N)$ for objects $M$, $N$ in $\mathcal{C}$, but am a little puzzled by some of the constructions I have seen, and the difference between $\text{Ext}$ for $\text{Mod}_A$ and for $\mathcal{C}$.
In III.2 of Hilton & Stammbach's A Course in Homological Algebra (2nd Ed.), we calculate $\text{Ext}(M,N)$ for $A$-modules $M$ and $N$ by choosing a projective presentation i.e. a short exact sequence $$0\to R\to P\to M\to 0,$$ where $P$ is projective. If we denote the morphism $R\to P$ by $\mu$ and the contravariant functor $D:=\text{Hom}_A(-,N)$, then we define $\text{Ext}_A(M,N):=\text{cok}(D\mu)$ and can check that this is essentially independent of the choice of projective presentation by constructing a natural isomorphism.
But how does this work for calculating $\text{Ext}$ within $\mathcal{C}$?