Double dual of a finitely generated Projective module over an artin algebra

Question

In Auslander’s Representation Theory of Artin Algebras, in chapter 2, the proposition 4.3 is stated that

(b) Let $\Lambda$ be an artin algebra, for each $P$ in ${\mathscr{P}}(\Lambda)$, the morphism $\phi_{P}: P\to P^{**}$ is an isomorphism.

where ${\mathscr{P}}(\Lambda)$ is the category of projective modules over $\Lambda$, and $P^{**}$ is the double dual of $P$, $Hom_{\Lambda}(Hom_{\Lambda}(P,\Lambda),\Lambda)$.

I am confused about the proof given in the book, how should I show it is an isomorphism using the additivity of $Hom_{\Lambda}(-,\Lambda)$?

Answer 1

The evaluation map determines an additive functor $\mathrm{ev}\colon\mathrm{id}\to(-)^{\ast\ast}$. Since it is an isomorphism for $\Lambda$, it induces (by naturality and additivity) an isomorphism for all $P\in\mathrm{add}(\Lambda)=\mathscr P(\Lambda)$.