Category of representations is Tannakian

Question

Tannakian categories are modeled on categories of linear representations. I am trying to learn about both subjects, but I cannot see why a category of representations should be a rigid tensor category. Specifically: for each object $M$ in a rigid tensor category $\mathcal{T}$, the axioms require that the dual object $M^\vee$ should satisfy the property that the functor $-\otimes M^\vee$ is left-adjoint to the functor $-\otimes M$. But why does this property hold for representations? If I have a morphism of vector spaces (representations) $A\otimes M^\vee \to B$, how do I get a morphism $A\to B \otimes M$?

Answer 1

This only works for finite-dimensional vector spaces. In that case the canonical map $M\to M^{\vee\vee}$, $x\mapsto \widehat{x}$, where $\widehat{x}$ is given by $\widehat{x}(\xi) = \xi(x)$ for $\xi\in M^\vee$, is an isomorphism.

Hence, we have $B\otimes M\cong B\otimes M^{\vee\vee} \cong \hom(M^\vee, B)$. But then we have natural isomorphisms $$ \hom(A\otimes M^\vee, B) \cong \hom(A, \hom(M^\vee, B)) \cong \hom(A, B\otimes M) $$ via the tensor-hom adjunction.