Let $(M,\otimes,1_M)$ be a monoidal category, and let $X$ 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 $\epsilon$ denotes evolution and $\eta$ the coevaluation). Let us assume that we have isomorphisms $l:M^* \to M$ and $r:{}^*M \to M$. 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 $$ (f_l \otimes \mathrm{id}) \circ \epsilon_l = (\mathrm{id} \otimes f_r) \circ \epsilon_r: M \otimes M \to 1_M. $$