Kerodon

Remark 2.2.4.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a lax functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. Then the identity constraints $\{ \epsilon _{X}: \operatorname{id}_{ F(X) } \Rightarrow F( \operatorname{id}_{X} ) \} _{X \in \operatorname{Ob}(\operatorname{\mathcal{C}}) }$ are uniquely determined by the other data of Definition 2.2.4.5. This follows from Proposition 2.1.5.4, applied to the lax monoidal functor $F_{X,X}: \underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X) \rightarrow \underline{\operatorname{End}}_{\operatorname{\mathcal{D}}}( F(X) )$ of Remark 2.2.4.7.