A right closed monoidal category $\mathcal{C}$ is one for which the functor $X \otimes -$ admits a right adjoint, for every $X \in \mathcal{C}$. As explained here

this gives an enrichement of $\mathcal{C}$ over itself. In this question

Rigid monoidal and closed monoidal categories

it says that rigid monoidal categories are closed monoidal. So what does the self-enrichment of a rigid monoidal category look like?

defines$[X,Y] = X^\ast \otimes Y$. The idea to keep in mind is that if you already had the closed monoidal structure, then you would have defined $X^\ast = [X,I]$. You would have a map $X^\ast \otimes Y = [X,I] \otimes [I,Y] \to [X,Y]$ internalizing the composition operation $Hom(X,I) \times Hom(I,Y)\to Hom(X,Y)$, and using rigidity you'd have shown this map to be an isomorphism. When setting up the closed structure, you run that picture backwards $\endgroup$