Corollary 2.3.2.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a strictly unitary lax functor. Then $F$ is a functor if and only if the induced map of simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \Rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$ carries thin $2$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ to thin $2$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\sigma $ be a $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, corresponding to a diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{\gamma } & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z } \]
in $\operatorname{\mathcal{C}}$. Let $\sigma '$ denote the image of $\sigma $ in $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$, corresponding to the diagram
\[ \xymatrix@R =50pt@C=50pt{ & F(Y) \ar [dr]^{F(g)} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^{\gamma '} & \\ F(X) \ar [ur]^{F(f)} \ar [rr]_{F(h)} & & F(Z) } \]
where $\gamma '$ is given by the (vertical) composition
\[ F(g) \circ F(f) \xRightarrow { \mu _{g,f} } F(g \circ f) \xRightarrow {F(\gamma )} F(h). \]
Since $\sigma $ is thin, the $2$-morphism $\gamma $ is an isomorphism (Theorem 2.3.2.5). It follows that $\sigma '$ is thin if and only if $\mu _{g,f}$ is an isomorphism. In particular, the strictly unitary lax functor $F$ preserves thin $2$-simplices if and only if $\mu _{g,f}$ is an isomorphism for every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ of $\operatorname{\mathcal{C}}$: that is, if and only if $F$ is a functor. $\square$