Remark 3.3.3.21 (Functoriality). Let $u: X \rightarrow Y$ be a morphism of braced simplicial sets. Then $u$ induces a morphism between their subdivisions
which can be identified with a functor $U: \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}} \rightarrow \operatorname{{\bf \Delta }}_{Y}^{\mathrm{nd}}$ (Proposition 1.3.3.1). If $u$ carries nondegenerate simplices of $X$ to nondegenerate simplices of $Y$, then the functor $U$ is easy to describe: it is given on objects by the formula $U( [n], \sigma ) = ( [n], u(\sigma ) )$. More generally, $U$ carries an object $([n], \sigma ) \in \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}$ to an object $([m], \tau ) \in \operatorname{{\bf \Delta }}_{Y}^{\mathrm{nd}}$, characterized by the requirement that $u(\sigma )$ factors as a composition $\Delta ^{n} \twoheadrightarrow \Delta ^{m} \xrightarrow {\tau } Y$ (see Proposition 1.1.3.8).