Terminal object

Definition

Let $\mathcal C$ be a category. A terminal object is an object $1$ such that for every object $X$ there is exactly one morphism

Any two terminal objects are uniquely isomorphic, so notation such as $1$ refers to a choice that is immaterial up to unique isomorphism.

For example, a one-point set is terminal among sets, and a one-point space is terminal among topological spaces. In a category of objects over a fixed base $S$, the identity $S\to S$ is terminal. Consequently, on a site with terminal object $1$, a covering family $\{U_i\to 1\}$ expresses a global object as being covered by local pieces.