How to show that a certain module is injective over an endomorphism algebra?

Question

Let $A$ be a self-injective Artin algebra and $M\in\ \mathfrak{mod}\ A$ with the property $\mathfrak{add}\ _AA = \mathfrak{add}\ M$. Let $I$ be a finitely generated injective $A$-module.

Why is then $\text{Hom}_A(M,I)$ injective as module over the algebra $C:=\text{End}_A(M)$ ?

I know that $\text{Hom}_A(M,I)$ is projective as module over the algebra $C$, if that helps...

For $X\in\mathfrak{mod}\ A$ we have: $\mathfrak{add}\ X$ denotes the full subcategory of $\mathfrak{mod}\ A$ whose objects are direct summands of finite direct sums of copies of $X$.

Thanks for the help!

Answer 1

Since $\mathfrak{add}\ _AA = \mathfrak{add}\ M$, we have that $A$ and $C=\text{End}_A(M)$ are Morita-equivalent, and that the functor $\text{Hom}_A(M,-): \text{Mod }A \to \text{Mod }C$ is an equivalence of categories. Since equivalences of categories preserve injective objects, the property is proved.