3 added particular case of what was last isomorphism ; made small stylistic changes in first sentence.

People

For those interested in general statementsmight like to notice the following results stating that , here is a summary of assumptions under which the canonical morphisms of $A$-modules below are isomorphismsunder the stated assumptions.:

If $P$ is finitely generated projective: $$P\stackrel{\sim }{\to} P^{\vee }{^\vee} \quad$$

A module $P$ is finitely generated projective iff the following canonical map below is an isomorphism

$$\quad P\otimes P^\vee \otimes P \stackrel{\sim }{\to} End(P)$$

If $P$ or $P'$ is finitely generated projective

$$P^\vee \otimes P' \stackrel{\sim }{\to} Hom(P,P')$$

If both $P$ and $P'$ or both $P$ and $M$ or both $P'$ and $M'$are finitely generated projective $$Hom(P,M) \otimes Hom(P',M') \stackrel{\sim }{\to} Hom(P\otimes P',M\otimes M')$$

In particular for $P$ or $P'$ finitely generated projective

$$P^\vee \otimes P'^\vee \stackrel{\sim }{\to} (P \otimes P')^\vee$$

(this is mephisto's answer follows from to the case $M=M'=A$)OP's question)

2 Changed hypotheses in order to have more general statements

People interested in general statements might like to notice the following results stating that the canonical morphisms of $A$-modules below are isomorphisms under the stated assumptions.

If $P$ is finitely generated projective: $$P\stackrel{\sim }{\to} P^{\vee }{^\vee} \quad and quad$$

A module $P$ is finitely generated projective iff the canonical map below is an isomorphism

$$\quad P\otimes P^\vee \stackrel{\sim }{\to} End(P)$$End(P) $$If moreover M is an arbitrary P or A- moduleP' is finitely generated projective$$ P\otimes M P^\vee \otimes P' \stackrel{\sim }{\to} Hom(P^\vee,MHom(P,P') $$If moreover both P and P' is a finitely generated projective or both P and M' an arbitrary M or both A-moduleP' and M'are finitely generated projective$$Hom(P,M) \otimes Hom(P',M') \stackrel{\sim }{\to} Hom(P\otimes P',M\otimes M') $$(The OP's question is, of course, mephisto's answer follows from the case M=M'=A) 1 People interested in general statements might like to notice the following results stating that the canonical morphisms of A-modules below are isomorphisms under the stated assumptions. If P is finitely generated projective:$$P\stackrel{\sim }{\to} P^{\vee }{^\vee} \quad and \quad P\otimes P^\vee \stackrel{\sim }{\to} End(P)$$If moreover M is an arbitrary A- module$$ P\otimes M \stackrel{\sim }{\to} Hom(P^\vee,M) $$If moreover P' is a finitely generated projective and M' an arbitrary A-module$$Hom(P,M) \otimes Hom(P',M') \stackrel{\sim }{\to} Hom(P\otimes P',M\otimes M') 

(The OP's question is, of course, the case $M=M'=A$)