Kerodon

Exercise 2.2.4.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a strict functor. Show that, for each morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the functor $F_{X,Y}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ carries the left and right unit constraints $\lambda _{f}: \operatorname{id}_{Y} \circ f \xRightarrow {\sim } f$ and $\rho _{f}: f \circ \operatorname{id}_{X} \xRightarrow {\sim } f$ to $\lambda _{F(f) }$ and $\rho _{F(f)}$, respectively (see Construction 2.2.1.11).