Remark 2.1.2.18. Let $\operatorname{\mathcal{C}}$ be a monoidal category. Then the left and right unit constraints $\lambda _{X}: \mathbf{1} \otimes X \xrightarrow {\sim } X$ and $\rho _{X}: X \otimes \mathbf{1} \xrightarrow {\sim } X$ depend functorially on $X$. In other words, for every morphism $f: X \rightarrow Y$, the diagram
\[ \xymatrix@R =50pt@C=50pt{ \mathbf{1} \otimes X \ar [d]^-{ \operatorname{id}_{ \mathbf{1}} \otimes f } \ar [r]^-{ \lambda _{X} }_{\sim } & X \ar [d]^{ f } & X \otimes \mathbf{1} \ar [l]_-{ \rho _ X}^{\sim } \ar [d]^{f \otimes \operatorname{id}_{ \mathbf{1} }}\\ \mathbf{1} \otimes Y \ar [r]^-{ \lambda _{Y} }_{\sim } & Y & Y \otimes \mathbf{1} \ar [l]_-{ \rho _ Y}^{\sim } } \]
is commutative.