The Stacks project

Lemma 4.43.9. 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. To show these arrows are mutually inverse, consider a map $a : Z' \to Z \otimes Y$. We have to show that

\[ Z' \xrightarrow {\text{id}_{Z'} \otimes \eta } Z' \otimes X \otimes Y \xrightarrow {a \otimes \text{id}_{X \otimes Y}} Z \otimes Y \otimes X \otimes Y \xrightarrow {\text{id}_ Z \otimes \epsilon \otimes \text{id}_ Y} Z \otimes Y \]

is equal to $a$. The composition of the first two arrows equals $(\text{id}_{Z \otimes Y} \otimes \eta ) \circ a : Z' \to Z \otimes Y \to Z \otimes Y \otimes X \otimes Y$. Then the composition of $\text{id}_{Z \otimes Y} \otimes \eta $ and $\text{id}_ Z \otimes \epsilon \otimes \text{id}_ Y$ equals the identity by definition of the dual. Similarly for the other composition. We omit the proof of the second equality. $\square$

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