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Notation Let $X$ be a simplicial set. It follows immediately from the definitions that if there exists a simplicial set $Y$ and a morphism $u: X \rightarrow \operatorname{Ex}(Y)$ which exhibits $Y$ as a subdivision of $X$, then the simplicial set $Y$ (and the morphism $u$) are uniquely determined up to isomorphism and depend functorially on $X$. To emphasize this dependence, we will denote $Y$ by $\operatorname{Sd}(X)$ and refer to it as the subdivision of $X$.