lemma. universal arrow via initial/terminal object of comma category [kostecki2011introduction, 3.6] [tt-001B]

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A universal arrow from \(X \in {\cal D}\) to \(\mathscr {F} : {\cal C} \to {\cal D}\) is an initial object in the comma category \(X \downarrow \mathscr {F}\).

Conversely, a (co)universal arrow from \(\mathscr {F} : {\cal C} \to {\cal D}\) to \(X \in {\cal D}\) is a terminal object in the comma category \(\mathscr {F} \downarrow X\).