In Method of Homological Algebra by Gelfand and Manin (Exercise 2.2.3).
How are $\mathrm{Hom}_A(P,X)$ and $\mathrm{Hom}_B(P^*,Y)\,$ regarded as a $B$-module and $A$-module respectively?
2026-04-03 16:22:23.1775233343
Modules in Morita Equivalence
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$\operatorname{Hom}_A(P,X)$ is a $B=\operatorname{End}_AP$-module via $f\cdot b:= f\circ b$ whild $\operatorname{Hom}_B(P^*,Y)$ is an $A$-module via $(f\cdot a)(p):=f(ap)$, where for $ap$ you use the left $A$-module structure on $P^*$. I leave it as an exercise for you to check that this defines right module structures.