Indexed Family

Definition

An indexed family is a functor $F$ from a discrete category $\mathcal{I}$ to a category $\mathcal{C}$:

$$F:\mathcal{I}\to \mathcal{C}$$

Since $\mathcal{I}$ is discrete (only identity morphisms), this data is exactly a collection of objects $\{X_i{\}}_{i\in \text{Ob}(\mathcal{I})}$.