The Stacks project

Example 14.3.5. Suppose that $Y\to X$ is a morphism of $\mathcal{C}$ such that all the fibred products $Y \times _ X Y \times _ X \ldots \times _ X Y$ exist. Then we set $U_ n$ equal to the $(n + 1)$-fold fibre product, and we let $\varphi : [n] \to [m]$ correspond to the map (on “coordinates”) $(y_0, \ldots , y_ m) \mapsto (y_{\varphi (0)}, \ldots , y_{\varphi (n)})$. In other words, the map $U_0 = Y \to U_1 = Y \times _ X Y$ is the diagonal map. The two maps $U_1 = Y \times _ X Y \to U_0 = Y$ are the projection maps.