The Stacks project

Definition 59.10.1. Let $\mathcal{C}$ be a category. A family of morphisms with fixed target $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I}$ is the data of

1. an object $U \in \mathcal{C}$,

2. a set $I$ (possibly empty), and

3. for all $i\in I$, a morphism $\varphi _ i : U_ i \to U$ of $\mathcal{C}$ with target $U$.