$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 5.4.7.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $(\infty ,2)$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $F$ is a functor: that is, it carries thin $2$-simplices of $\operatorname{\mathcal{C}}$ to thin $2$-simplices of $\operatorname{\mathcal{D}}$.
- $(2)$
For every pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ of $\operatorname{\mathcal{C}}$, there exists a thin $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ with $d^{2}_0(\sigma ) = g$ and $d^{2}_2(\sigma ) = f$, as indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr] & & Z, } \]
such that $F(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$.
Proof.
The implication $(1) \Rightarrow (2)$ is immediate. To prove the converse, let $T$ be the collection of all $2$-simplices of $\operatorname{\mathcal{C}}$ for which $F(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$. Since the collection of thin $2$-simplices of $\operatorname{\mathcal{D}}$ has the four-out-of-five property (Proposition 5.4.6.11), it follows that $T$ also has the four-out-of-five property (Remark 5.4.6.10). Since the collection of thin $2$-simplices of $\operatorname{\mathcal{D}}$ has the inner exchange property (Proposition 5.4.5.10), $T$ has the inner exchange property (Remark 5.4.5.9). Since $\operatorname{\mathcal{D}}$ is an $(\infty ,2)$-category, every degenerate $2$-simplex of $\operatorname{\mathcal{D}}$ is thin, so every degenerate $2$-simplex of $\operatorname{\mathcal{C}}$ belongs to $T$. If condition $(2)$ is satisfied, then Proposition 5.4.6.14 guarantees that every thin $2$-simplex of $\operatorname{\mathcal{C}}$ belongs to $T$, so that $F$ is a functor.
$\square$