Lemma 4.43.6. Let $\mathcal{C}$ be a monoidal category. If $Y$ is a left dual to $X$, then
\[ \mathop{\mathrm{Mor}}\nolimits (Z' \otimes X, Z) = \mathop{\mathrm{Mor}}\nolimits (Z', Z \otimes Y) \quad \text{and}\quad \mathop{\mathrm{Mor}}\nolimits (Y \otimes Z', Z) = \mathop{\mathrm{Mor}}\nolimits (Z', X \otimes Z) \]
functorially in $Z$ and $Z'$.
Proof.
Consider the maps
\[ \mathop{\mathrm{Mor}}\nolimits (Z' \otimes X, Z) \to \mathop{\mathrm{Mor}}\nolimits (Z' \otimes X \otimes Y, Z \otimes Y) \to \mathop{\mathrm{Mor}}\nolimits (Z', Z \otimes Y) \]
where we use $\eta $ in the second arrow and the sequence of maps
\[ \mathop{\mathrm{Mor}}\nolimits (Z', Z \otimes Y) \to \mathop{\mathrm{Mor}}\nolimits (Z' \otimes X, Z \otimes Y \otimes X) \to \mathop{\mathrm{Mor}}\nolimits (Z' \otimes X, Z) \]
where we use $\epsilon $ in the second arrow. A straightforward calculation using the properties of $\eta $ and $\epsilon $ shows that the compositions of these are mutually inverse. Similarly for the other equality.
$\square$
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