The Stacks project

Definition 4.12.1. Let $\mathcal{C}$ be a category.

1. An object $x$ of the category $\mathcal{C}$ is called an initial object if for every object $y$ of $\mathcal{C}$ there is exactly one morphism $x \to y$.

2. An object $x$ of the category $\mathcal{C}$ is called a final object if for every object $y$ of $\mathcal{C}$ there is exactly one morphism $y \to x$.