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Proposition Let $X$ be a braced simplicial set. Then the morphism $f: X \rightarrow \operatorname{Ex}( \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}) )$ of Construction exhibits the nerve $\operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}} )$ as a subdivision of $X$, in the sense of Definition

Proof of Proposition Let $S$ be a semisimplicial set, and let $S^{+}$ denote the braced simplicial set given by Construction Applying Construction, we obtain a comparison map of simplicial sets $u_{S}: \operatorname{Sd}( S^{+} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}})$. We wish to show that $u_ S$ is an isomorphism for every semisimplicial set $S$. Note that the functor $S \mapsto \operatorname{Sd}( S^{+} )$ preserves colimits (since it is a left adjoint) and the functor $S \mapsto \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}})$ also preserves colimits (by Lemma Since every functor $S: \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ can be written as a colimit of representable functors (see ยง), we may assume without loss of generality that $S \simeq ( \Delta ^{n} )^{\mathrm{nd} }$ is the semisimplicial set represented by an object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj}}$. In this case, the desired comparison is immediate from the definition of subdivision (see Examples and $\square$