The Stacks project

Definition 7.6.1. Let $\mathcal{C}$ be a category, see Conventions, Section 2.3. A family of morphisms with fixed target in $\mathcal{C}$ is given by an object $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, a set $I$ and for each $i\in I$ a morphism $U_ i \to U$ of $\mathcal{C}$ with target $U$. We use the notation $\{ U_ i \to U\} _{i\in I}$ to indicate this.