corollary. a representation is a universal element [leinster2016basic, 4.3.2] [tt-0035]

✍️source

Let \({\cal C}\) be a locally small category and \(\mathscr {F} : {\cal C}^{op} \to \mathbf {Set}\). Then a representation of \(\mathscr {F}\) consists of a pair \((X, u)\) such that the diagram commutes.