Let $(\mathcal{A},\otimes,1_{\mathcal{A}})$ be a monoidal category, and let $M$ be a rigid object of $M$, with left and right dual respectively denoted by
$$
\Big\{M^*,~~~ \epsilon_l:M^* \otimes M \to 1_{\mathcal{A}},~~~ \eta_l:1_{\mathcal{A}} \to M \otimes M^*\Big\}
$$
and
$$
\Big\{{}^*M, ~~~\epsilon_r: M \otimes {}^*M \to 1_{\mathcal{A}},~~~\eta_r: 1_{\mathcal{A}} \to {}^*M \otimes M\Big\}
$$ 
respectively (just as in [Wikipedia][1] $\epsilon$ denotes **evaluation** and $\eta$ the **coevaluation**). Let's assume that we have isomorphisms $f_l:M^* \to M$ and $f_r:{}^*M \to M$. Then do we have the identity 
$$
(\mathrm{id} \otimes f_l) \circ \eta_l(1) = (f_r \otimes \mathrm{id}) \circ \eta_r(1)?
$$
This works for finite dimensional vector spaces, so I have guessed that it works in general. Also, do we have the identities 
$$
(f_l \otimes \mathrm{id}) \circ \epsilon_l = (\mathrm{id} \otimes f_r) \circ \epsilon_r: M \otimes M \to 1_M?
$$


  [1]: https://en.wikipedia.org/wiki/Rigid_category