Let $A, B$ be $R$-mods. I am trying to prove that if, for all injective $R$-mods $Q$, $Hom(B, Q) \rightarrow Hom(A, Q)$ surjective $\implies A \rightarrow B$ is injective. (Hint: R-mod has enough injectives)
At first I tried to avoid the given hint by using the fact that $Hom(_, Q) is exact but that led me no where. I am not sure how I can utilize the hint.
Hint already given to you: There exists an injective object $Q$ and a monomorphism $A\to Q$.
Another hint: if $g\circ f$ is monomorphism, then so is $f$.