Example 2.2.4.11 (008M)

Example 2.2.4.11. 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 (in the sense of Definition 2.2.4.1). Then we can regard $F$ as a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ (in the sense of Definition 2.2.4.5) by taking the identity and composition constraints

\[ \epsilon _ X: \operatorname{id}_{F(X)} \Rightarrow F( \operatorname{id}_ X) \quad \quad \mu _{g,f}: F(g) \circ F(f) \Rightarrow F( g \circ f) \]

to be the identity maps (note that in this case, conditions $(a)$, $(b)$, and $(c)$ of Definition 2.2.4.5 reduce to conditions $(3)$ and $(4)$ of Definition 2.2.4.1). Conversely, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a lax functor having the property that each of the identity and composition constraints $\epsilon _{X}$ and $\mu _{g,f}$ is an identity $2$-morphism of $\operatorname{\mathcal{D}}$, then we can regard $F$ as a strict $2$-functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. We therefore have inclusions

\[ \{ \text{Strict functors $F:\operatorname{\mathcal{C}}\to \operatorname{\mathcal{D}}$} \} \subseteq \{ \text{Functors $F: \operatorname{\mathcal{C}}\to \operatorname{\mathcal{D}}$} \} \subseteq \{ \text{Lax functors $F: \operatorname{\mathcal{C}}\to \operatorname{\mathcal{D}}$} \} . \]

In general, neither of these inclusions is reversible.