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Remark Let $X$ be a simplicial set. Then the construction $Y \mapsto X \star Y$ determines a functor

\[ \operatorname{Set_{\Delta }}\rightarrow (\operatorname{Set_{\Delta }})_{X/} \]

which preserves small colimits. It follows that the composite functor

\[ \operatorname{Set_{\Delta }}\rightarrow (\operatorname{Set_{\Delta }})_{X/} \rightarrow \operatorname{Set_{\Delta }}\quad \quad Y \mapsto X \star Y \]

preserves filtered colimits and pushouts. Beware that it does not preserve colimits in general (for example, it carries the initial object $\emptyset \in \operatorname{Set_{\Delta }}$ to the simplicial set $X$, which need not be initial).