Kerodon

Definition 2.2.4.17. 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. We say that $F$ is unitary if, for every object $X \in \operatorname{\mathcal{C}}$, the identity constraint $\epsilon _{X}: \operatorname{id}_{ F(X)} \Rightarrow F( \operatorname{id}_ X)$ is an invertible $2$-morphism of $\operatorname{\mathcal{D}}$. We say that $F$ is strictly unitary if, for every object $X \in \operatorname{\mathcal{C}}$, we have an equality $\operatorname{id}_{F(X)} = F( \operatorname{id}_ X)$ and the identity constraint $\epsilon _{X}$ is the identity $2$-morphism from $\operatorname{id}_{F(X)}$ to itself.