Item 2 (002G) — The Clowder Project

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Associativity. Given a diagram

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

\[ \webleft (A\mathbin {\textstyle \coprod _{X}}B\webright )\mathbin {\textstyle \coprod _{Y}}C\cong \webleft (A\mathbin {\textstyle \coprod _{X}}B\webright )\mathbin {\textstyle \coprod _{B}}\webleft (B\mathbin {\textstyle \coprod _{Y}}C\webright ) \cong A\mathbin {\textstyle \coprod _{X}}\webleft (B\mathbin {\textstyle \coprod _{Y}}C\webright ), \]

where these pullbacks are built as in the diagrams

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