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Construction Let $f: A \hookrightarrow A'$ and $g: B \hookrightarrow B'$ be monomorphisms of simplicial sets. Using Remark, we see that the induced maps

\[ A \star B' \xrightarrow { f \star \operatorname{id}_{B'} } A' \star B' \xleftarrow { \operatorname{id}_{A'} \star g } A' \star B \]

are also monomorphisms. Moreover, the intersection of their images is the image of the monomorphism $(f \star g): A \star B \hookrightarrow A' \star B'$. We therefore obtain a monomorphism of simplicial sets

\[ (A \star B') \coprod _{ (A \star B) } (A' \star B) \hookrightarrow A' \star B', \]

which we will refer to as the pushout-join of $f$ and $g$.