Proposition 13

Let $B$ be a ring, $F$ an injective cogenerator $B$-module, $P$ a $(B,A)$-bimodule. Suppose that $P$ is flat over $A$ and such that $P\otimes_A S\ne 0$ for every simple $A$-module $S$ (* that is, faithfully flat over $A$ in the sense of AC, I, §). Then the $A$-module $\operatorname{Hom}_B(P,F)$ is a cogenerator and injective.

Indeed, $\operatorname{Hom}_B(P,F)$ is injective by prop. 11. On the other hand, for every simple $A$-module $S$, $\operatorname{Hom}_A(S,\operatorname{Hom}_B(P,F))$ is isomorphic to $\operatorname{Hom}_B(P\otimes_A S,F)$, hence is nonzero since $P\otimes_A S\ne 0$ and the $B$-module $F$ is a cogenerator; the $A$-module $\operatorname{Hom}_B(P,F)$ is therefore a cogenerator by prop. 12.