In a rigid monoidal category, why is $W^*\otimes V^*$ a left dual of $V \otimes W$? Question: Let $\mathcal{C}$ be a monoidal category, $V,W$ in $\mathcal{C}$ are objects. Show that if $V, W$ have left duals $V^*, W^*$, respectively, then $V\otimes W$ has a left dual $W^* \otimes V^*$.
(Exercise 2.10.7b in Tensor Categories by Etingof, Gelaki, Nikshych, Ostrik (AMS page, author pdf))
Other notation: let $\mathbf{1}$ denote the unit in $\mathcal{C}$ and $l_V, r_V$ the left and right associators corresponding to $V \in \mathcal{C}$. Let $c_V, e_V$ denote the coevaluation and evaluation maps associated to $V$ (similarly for $W$). Let $a$ denote the associativity constraint and $1_V$ the identity morphism of $V$.
Note: As I was typing up the question, I think I figured out the answer. See below.
 A: My approach:  We want to produce a coevaluation map $c: \mathbf{1} \rightarrow (V \otimes W) \otimes (W^* \otimes V^*)$ and an evaluation map $e : (W^* \otimes V^*) \otimes (V \otimes W)  \rightarrow    \mathbf{1} $ such that the maps
$$r_{V\otimes W} \circ (1_{V\otimes W} \otimes e) \circ a_{V\otimes W, W^* \otimes V^*, V\otimes W} \circ c\otimes 1_{V\otimes W} \circ l_{V\otimes W}^{-1}$$
$$l_{W^*\otimes V^*} \circ (e\otimes 1_{W^*\otimes V^*} ) \circ a_{W^*\otimes V^*, V \otimes W, W^*\otimes V^*}^{-1} \circ 1_{W^*\otimes V^*}\otimes c \circ r_{W^*\otimes V^*}^{-1}$$
are the identity maps on $V \otimes W$ and $W^* \otimes V^*$, respectively. Naturally one would want to use the maps $e_V, c_V, e_W, c_W$, so I suggest the following definition of $e$, suppressing the associativity constraint:
$$  e = W^* \otimes V^* \otimes V \otimes W  \xrightarrow{1_W^* \otimes e_V \otimes 1_V^*} W^*\otimes \mathbf{1} \otimes W \cong W^* \otimes W \xrightarrow{ev_W} \mathbf{1}$$
and similarly
$$  c = \mathbf{1} \xrightarrow{c_V} V^*\otimes V \cong (V^* \otimes 1) \otimes V \xrightarrow{1_{V^*} \otimes e_W \otimes 1_V } V \otimes W \otimes W^* \otimes V^*$$
It can be checked that these give $W^* \otimes V^*$ the structure of a left dual by verifying the definition. Then, since all left duals are unique up to unique isomorphism, it follows that $(V \otimes W)^* \cong W^*\otimes V^*$.
