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Remark Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be strictly unitary $2$-categories (Definition Then a strictly unitary lax functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is given by the following data:

  • For each object $X \in \operatorname{\mathcal{C}}$, an object $F(X) \in \operatorname{\mathcal{D}}$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, a functor of ordinary categories

    \[ F_{X,Y}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ). \]
  • For every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in $\operatorname{\mathcal{C}}$, a composition constraint $\mu _{g,f}: F(g) \circ F(f) \Rightarrow F(g \circ f)$, depending functorially on $f$ and $g$.

This data must be required to satisfy axiom $(c)$ of Definition, together with the identities $F( \operatorname{id}_ X) = \operatorname{id}_{F(X)}$ for each object $X \in \operatorname{\mathcal{C}}$ and $\mu _{\operatorname{id}_ Y, f} = \operatorname{id}_{ F(f)} = \mu _{f, \operatorname{id}_ X}$ for each $1$-morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$.