For every object $r$ in a category $C$, we can consider the "hom-in" functor that is dual to the hom-out functor, that is the functor ${\mathrm{Hom}}_{C}(\beta \x88\x92,r)$. In general this is a functor from ${C}^{\text{op}}$ to $\mathbf{S}\mathbf{e}\mathbf{t}$, although just as with the hom-out functor in the case of $R$-modules we can consider it a functor with values in the category of abelian groups.

We now analyze the properties of this functor, in parallel with those of the hom-out functor.

The hom-in functor and direct products, sums

How does the hom-in functor interact with direct products? At first glance, it seems somewhat differently than the hom-out functor:

The hom-out functor exchanges direct sums for direct products

Suppose $D$ is an $R$-module and $\{{N}_{a}\beta \x88\pounds a\beta \x88\x88A\}$ is a family of $R$-modules. There is an $R$-module isomorphism

given by sending a morphism $f:\underset{a\beta \x88\x88A}{\beta \xa8\x81}{N}_{a}\beta \x86\x92D$ to the family of morphisms $(f\beta \x88\x98{i}_{a}{)}_{a\beta \x88\x88A}$, where $i}_{a}:{N}_{a}\beta \x86\x92\underset{{a}^{\beta \x80\xb2}\beta \x88\x88A}{\beta \xa8\x81}{N}_{{a}^{\beta \x80\xb2}$ is the canonical injection.

Recall that for finite families the direct product and direct sum constructions are isomorphic, so in that case we can replace the direct product with a direct sum. Because of this, we might sometimes say that "the hom-in functor commutes with finite direct sums." The above result is a more honest picture of the general case, though.

This is actually directly analogous to the property for the hom-out functor, once we recall that the hom-in functor is a contravariant functor; i.e., ${\mathrm{Hom}}_{R}(\beta \x88\x92,D):(R\beta \x88\x92\mathbf{\text{Mod}}{)}^{\text{op}}\beta \x86\x92\mathbf{\text{Ab}}$. Note that direct products in the category $(R\beta \x88\x92\mathbf{\text{Mod}}{)}^{\text{op}}$ correspond to direct sums in the category $R\beta \x88\x92\mathbf{\text{Mod}}$. So the above isomorphism can be viewed as the statement that the functor ${\mathrm{Hom}}_{R}(\beta \x88\x92,D):(R\beta \x88\x92\mathbf{\text{Mod}}{)}^{\text{op}}\beta \x86\x92\mathbf{\text{Ab}}$ commutes with direct products in the domain and codomain categories. This is the identical property enjoyed by the functor ${\mathrm{Hom}}_{R}(D,\beta \x88\x92):R\beta \x88\x92\mathbf{\text{Mod}}\beta \x86\x92\mathbf{\text{Ab}}$.

Notice that the $0$ on the far right of the sequence is gone! We have lost the "right end" of our exact sequence. Because of the above property, we say that the functor ${\mathrm{Hom}}_{R}(\beta \x88\x92,D):R\beta \x88\x92\mathbf{\text{Mod}}\beta \x86\x92\mathbf{\text{Ab}}$ is left exact.

Is it ever the case that the functor ${\mathrm{Hom}}_{R}(\beta \x88\x92,D)$ is exact, in other words sends short exact sequences to short exact sequences?

Definition of an injective module

An $R$-module $Q$ is injective if for every short exact sequence of $R$-modules