The Stacks project

Remark 90.21.5. Hence a representable groupoid in functors on $\mathcal{C}$ is given by objects $U$ and $R$ of $\mathcal{C}$ and morphisms $s, t : U \to R$ and $c : R \to R \amalg _{s, U, t} R$ such that $(\underline{U}, \underline{R}, s, t, c)$ satisfies the condition of Definition 90.21.1. The reason for requiring the existence of the pushout $R \amalg _{s, U, t} R$ is so that the composition morphism $c$ is defined at the level of morphisms in $\mathcal{C}$. This requirement will always be satisfied below when we consider representable groupoids in functors on $\widehat{\mathcal{C}}_\Lambda $, since by Lemma 90.4.3 the category $\widehat{\mathcal{C}}_\Lambda $ admits pushouts.