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

Enrichments vs Internal homs

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?

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

Enrichments vs Internal homs

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?