Let $(M,\otimes,1_M)$ be a monoidal category, and let $M$ be a rigid object of $M$, with left and right dual respectively denoted by $(M^*,\epsilon_l,\eta_l)$ and $({}^*M,\epsilon_r,\eta_r)$ respectively (just as in [Wikipedia][1] $\epsilon$ denotes evaluation and $\eta$ the coevaluation). Let us 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