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
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 natural maps.