In "Catégories Tannakiennes" by Savedra Rivano (under A. Grothendieck supervision) at pag.78 he define a rigid category $\mathscr{C}$ as a monoidal symmetrical closed such that the natural morphisms $[X, X'] \otimes [Y, Y'] \to [X \otimes Y, X' \otimes Y' ]$ and $\theta: X \mapsto [[X, I], I] $ are isomorphisms.

The first is adjoint to $[X, X'] \otimes [Y, Y'] \otimes X \otimes Y \cong [X, X'] \otimes X \otimes [Y, Y'] \otimes Y \xrightarrow{t \otimes t} X' \otimes Y' $

the latter is adjoint to $X \otimes [X, I] \cong [X, I] \otimes X \xrightarrow{t} I$.

where $t_{A, B}: [A, B] \otimes B \to A$, $u_{X, Y}: X \to [Y, X \otimes Y]$ the co-unity and unity of the adjunction $(A \otimes B, C) \cong (A, [B, C])$.

Let us write $[X, I] := \widehat{X}$.

Follow the natural isomorphisms:

1) $[X, Y] \cong \hat{X} \otimes Y $

2) $([X, Y])^\wedge \cong [\widehat{Y}, \widehat{X}] $

Now, in mathematical literature are studied many kinds of specialized monoidal categories: compact closed, tortile, autonomous, rigid (but a different definition) ecc. ecc. and many work about coherence questions about.

**I ask**: this example of "rigid category" (of Savedra Rivano) is a particular case of some well studied type of monoidal categories?

A more simple question: let $\tau_X: I\xrightarrow{j} [X, X]\cong \widehat{X}\otimes X$.

**I ask**: (in a rigid category) is the morphisms $I\xrightarrow{\tau_X}\widehat{X}\otimes X \xrightarrow{s}
X\otimes \widehat{X} \xrightarrow{\theta \otimes 1} \widehat{\widehat{X}}\otimes \widehat{X} $ equal to $\tau_{\widehat{X}}: I\to \widehat{\widehat{X}}\otimes \widehat{X}$ ?