$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Notation Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. To supply a lax $2$-functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, one must specify not only the values of $F$ on objects, $1$-morphisms, and $2$-morphisms of $\operatorname{\mathcal{C}}$, but also the identity and composition constraints

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

In situations where we need to consider more than one lax functor at a time, we will denote these $2$-morphisms by $\epsilon _{X}^{F}$ and $\mu _{g,f}^{F}$ (to avoid ambiguity).