Given a locally small category $\mathcal{C}$, Wikipedia defines the Hom functors as
At my lectures (for the more specific case of $\mathcal{C}=\operatorname{R-Mod}$ -- the category of R-modules for some ring R) the functor $Hom_R(-, M)$ was defined as a functor between $\operatorname{R-Mod}^{op}\rightarrow \operatorname{AbGrp}$.
- I think I understand why the codomain is $\operatorname{AbGrp}$, but I don't understand why the domain is the opposite category $\operatorname{R-Mod}^{op}$? How is it possible that Wikipedia gives the domain to be $\mathcal{C}$?
- Also, my notes claim that it follows from the left-exactness of the Hom-functor $Hom_\mathcal{A}(M, -): \mathcal{A} \rightarrow \operatorname{AbGrp}$ (here $\mathcal{A}$ is an arbitrary Abelian category) that $Hom_R(-, M): \operatorname{R-Mod}^{op} \rightarrow \operatorname{AbGrp}$ is also left exact. Could you explain why this is a corollary of the left-exactness of $Hom_\mathcal{A}(M,-): \mathcal{A}\rightarrow\operatorname{AbGrp}$?