definition. universal arrow [kostecki2011introduction, 3.4, 3.5] [tt-001A]

✍️source

A universal arrow from \(X \in {\cal D}\) to \(\mathscr {F} : {\cal C} \to {\cal D}\) is a unique pair \((O , u)\) that makes the diagram commute for any \(f \in {\cal D}\).

Conversely, a (co)universal arrow from \(\mathscr {F} : {\cal C} \to {\cal D}\) to \(X \in {\cal D}\) is a unique pair \((O , u)\) that makes the diagram commute for any \(f \in {\cal D}\).