Item 1 (00BS) — The Clowder Project

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*To be replaced with Linus Romer's Elemaints when it is released.

Associativity. We have isomorphisms of pointed sets

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

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

in $\mathsf{Sets}_{*}$.

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