## 4.12 Initial and final objects

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

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 $x \to y$.

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 $y \to x$.

In the category of sets the empty set $\emptyset $ is an initial object, and in fact the only initial object. Also, any *singleton*, i.e., a set with one element, is a final object (so it is not unique).

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