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](https://bookstore.ams.org/surv-205/), [author pdf](http://www-math.mit.edu/%7Eetingof/egnobookfinal.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.



In a rigid monoidal category, why is $W^*\otimes V^*$ a left dual of $V \otimes W$?
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 $(V \otimes W)^\* \cong W^\*\otimes V^\*$.


