Example 2.2.5.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be ordinary categories, which we regard as $2$-categories having only identity $2$-morphisms (see Example 2.2.0.6). Then every lax functor of $2$-categories from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is automatically strict (Example 2.2.4.14), and can be identified with a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ in the usual sense. In other words, we can view Example 2.2.0.6 as supplying fully faithful embeddings (of ordinary categories)
\[ \operatorname{Cat}\hookrightarrow \operatorname{2Cat}_{\operatorname{Str}} \quad \quad \operatorname{Cat}\hookrightarrow \operatorname{2Cat}\quad \quad \operatorname{Cat}\hookrightarrow \operatorname{2Cat}_{\operatorname{Lax}} \quad \quad \operatorname{Cat}\hookrightarrow \operatorname{2Cat}_{\operatorname{ULax}}. \]