Kerodon

Example 2.2.5.4. Let $\operatorname{\mathcal{C}}$ be a $2$-category. We let $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be the strict functor which carries every object, $1$-morphism, and $2$-morphism of $\operatorname{\mathcal{C}}$ to itself. We will refer to $\operatorname{id}_{\operatorname{\mathcal{C}}}$ as the identity functor on $\operatorname{\mathcal{C}}$. Note that it is both a left and right unit for the composition of lax functors given in Construction 2.2.5.1.