Item 2 (00BJ) — The Clowder Project

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Associativity. Given a diagram

in $\mathsf{Sets}$, we have isomorphisms of pointed sets

\[ \webleft (X\mathbin {\textstyle \coprod _{W}}Y\webright )\mathbin {\textstyle \coprod _{V}}Z\cong \webleft (X\mathbin {\textstyle \coprod _{W}}Y\webright )\mathbin {\textstyle \coprod _{Y}}\webleft (Y\mathbin {\textstyle \coprod _{V}}Z\webright ) \cong X\mathbin {\textstyle \coprod _{W}}\webleft (Y\mathbin {\textstyle \coprod _{V}}Z\webright ), \]

where these pullbacks are built as in the diagrams

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