Remark 4.5.8.6 (Functoriality). The blunt join construction $(X,Y) \mapsto X \diamond Y$ determines a functor $\diamond : \operatorname{Set_{\Delta }}\times \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$. Moreover:
For fixed $X$, the functor
\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad Y \mapsto X \diamond Y \]preserves monomorphisms, filtered colimits and pushout diagrams.
For fixed $Y$, the functor
\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad X \mapsto X \diamond Y \]preserves monomorphisms, filtered colimits, and pushout diagrams.