Remark 2.2.1.12. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. For every $1$-morphism $f: X \rightarrow X$ in $\operatorname{\mathcal{C}}$, the left and right unit constraints
\[ \lambda _{f}: \operatorname{id}_{X} \circ f \xRightarrow {\sim } f \quad \quad \rho _{f}: f \circ \operatorname{id}_ X \xRightarrow {\sim } f \]
of Construction 2.2.1.11 coincide with the left and right unit constraints of Construction 2.1.2.17, applied to the monoidal category $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$ of Remark 2.2.1.7.