Remark. universal element [leinster2016basic, 4.3.2] [tt-0036]

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Pairs \((Y, y)\) with \(Y \in {\cal C}\) and \(y \in \mathscr {F}(Y)\) in Corollary [tt-0035] are sometimes called elements of the presheaf \(\mathscr {F}\).

Indeed, Lemma [tt-002N] (Yoneda) tells us that \(y\) amounts to a generalized element of \(\mathscr {F}\) of shape \(\mathscr {H}_Y\).

An element \(u\) satisfying condition in Corollary [tt-0035] is sometimes called a universal element of \(\mathscr {F}\). So, Corollary [tt-0035] says that a representation of a presheaf \(\mathscr {F}\) amounts to a universal element of \(\mathscr {F}\).