Example 14.5.5. Suppose that $X\to Y$ is a morphism of $C$ such that all the pushouts $Y\amalg _ X Y \amalg _ X \ldots \amalg _ X Y$ exist. Then we set $U_ n$ equal to the $(n + 1)$-fold pushout, and we let $\varphi : [n] \to [m]$ correspond to the map
\[ (y \text{ in }i\text{th component}) \mapsto (y \text{ in }\varphi (i)\text{th component}) \]
on “coordinates”. In other words, the map $U_1 = Y \amalg _ X Y \to U_0 = Y$ is the identity on each component. The two maps $U_0 = Y \to U_1 = Y \amalg _ X Y$ are the two coprojections.
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