Universal property

(See @lawvere2009conceptual page 226)
An universal property in a category $\mathcal{C}$ is a property that describes the nature of a particular object $S$ in terms of its relation to all objects $X$ in $\mathcal{C}$. Moreover, the nature of the relation of $S$ to the objects $X$ is described in terms of the maps of the category, more precisely, saying that there exists exactly one map satisfying the given condition.

They can be used to define the particular class of objects.

The simplest universal property is that of terminal object.