Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Exercise 2.1.2.20. Let $\operatorname{\mathcal{C}}$ be a monoidal category with unit object $\mathbf{1}$. Show that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the diagrams

\[ \xymatrix { X \otimes (Y \otimes \mathbf{1}) \ar [rr]^{\alpha _{X,Y,\mathbf{1}} } \ar [dr]_{\operatorname{id}_ X \otimes \rho _{Y}} & & (X \otimes Y) \otimes \mathbf{1} \ar [dl]^{ \rho _{ X \otimes Y} } \\ & X \otimes Y & } \]
\[ \xymatrix { \mathbf{1} \otimes (X \otimes Y) \ar [dr]_{\lambda _{X \otimes Y}} \ar [rr]^{ \alpha _{ \mathbf{1}, X, Y} } & & ( \mathbf{1} \otimes X) \otimes Y \ar [dl]^{ \lambda _ X \otimes \operatorname{id}_ Y} \\ & X \otimes Y & } \]

are commutative (for a more general statement, see Proposition 2.2.1.16).