The Stacks project

Definition 90.21.1. Let $\mathcal{C}$ be a category. The category of groupoids in functors on $\mathcal{C}$ is the category with the following objects and morphisms.

1. Objects: A groupoid in functors on $\mathcal{C}$ is a quintuple $(U, R, s, t, c)$ where $U, R : \mathcal{C} \to \textit{Sets}$ are functors and $s, t : R \to U$ and $c : R \times _{s, U, t} R \to R$ are morphisms with the following property: For any object $T$ of $\mathcal{C}$, the quintuple

   $$(U(T), R(T), s, t, c)$$

   is a groupoid category.

2. Morphisms: A morphism $(U, R, s, t, c) \to (U', R', s', t', c')$ of groupoids in functors on $\mathcal{C}$ consists of morphisms $U \to U'$ and $R \to R'$ with the following property: For any object $T$ of $\mathcal{C}$, the induced maps $U(T) \to U'(T)$ and $R(T) \to R'(T)$ define a functor between groupoid categories

   $$(U(T), R(T), s, t, c) \to (U'(T), R'(T), s', t', c').$$