Item 1 (00AV) — The Clowder Project

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Associativity. We have isomorphisms of pointed sets

\[ \underbrace{\text{Eq}\webleft (f\circ \text{eq}\webleft (g,h\webright ),g\circ \text{eq}\webleft (g,h\webright )\webright )}_{{}=\text{Eq}\webleft (f\circ \text{eq}\webleft (g,h\webright ),h\circ \text{eq}\webleft (g,h\webright )\webright )}\cong \text{Eq}\webleft (f,g,h\webright ) \cong \underbrace{\text{Eq}\webleft (f\circ \text{eq}\webleft (f,g\webright ),h\circ \text{eq}\webleft (f,g\webright )\webright )}_{{}=\text{Eq}\webleft (g\circ \text{eq}\webleft (f,g\webright ),h\circ \text{eq}\webleft (f,g\webright )\webright )}, \]

where $\text{Eq}\webleft (f,g,h\webright )$ is the limit of the diagram

in $\mathsf{Sets}_{*}$, being explicitly given by

\[ \text{Eq}\webleft (f,g,h\webright )\cong \webleft\{ a\in A\ \middle |\ f\webleft (a\webright )=g\webleft (a\webright )=h\webleft (a\webright )\webright\} . \]

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