Take the tour ×  Let $\mathcal C$ be a symmetric monoidal closed
category. This means that every functor $- \otimes B$ has a
right adjoint $[B, -]$. Let $I$ be the unit and let $\rho \colon
- \otimes I \to 1_{\mathcal C}$ be the right unitor. There are
isomorphisms $$ \mathcal C (C \otimes I, C ) \cong \mathcal (C ,
[I,C])$$ because there is an adjunction. If we let $i_C \colon C
\to [I,C]$ be the morphism corresponding to $\rho_C$ under the
adjunction, then $i_C$ should be an isomorphism. I am having
trouble proving this however.  If we let $\mathrm{ev}_{IC}
\colon [I,C] \otimes I \to C$ be the morphism corresponding to
the identity under the adjunction (i.e. the counit of the
adjunction $- \otimes I \dashv [I, - ]$ ) then the inverse for
$i_C$ is supposed to be given by the morphism  $$ [I,C]
\xrightarrow{\rho_{[I,C]}^{-1}} [I,C] \otimes I
\xrightarrow{\mathrm{ev}_{IC}} C. $$  I can see quite easily
that $\mathrm{ev}_{IC} \circ \rho_{[I,C]}^{-1} \circ i_c = 1 $
but I can't for the life of me figure out why