theorem. density [leinster2016basic, 6.2.17] [tt-004D]

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Let \({\cal C}\) be a small category and \(\mathscr {X}\) a presheaf on \({\cal C}\). Then \(\mathscr {X}\) is the colimit of the diagram \[ {\cal E}(\mathscr {X}) \xrightarrow {\mathscr {P}} \mathbf {{\cal C}} \xrightarrow {H_{\bullet }}\left [\mathbf {{\cal C}}^{op}, \text { Set }\right ] \] in \(\left [\mathbf {{\cal C}}^{op}, \mathbf {Set} \right ]\), i.e. \[\mathscr {X} \cong \lim \limits _{\rightarrow {\cal E}(\mathscr {X})}\left (\mathscr {P} \mathbin {\bullet } H_{\bullet } \right )\] where \({\cal E}(\mathscr {X})\) is the category of elements, \(\mathscr {P}\) the projection functor in it, and \(H_{\bullet }\) the (contravariant) Yoneda embedding.

This theorem states that every presheaf is a colimit of representables in a canonical way. It is secretly dual to the Yoneda lemma. This becomes apparent if one expresses both in suitably lofty categorical language (that of ends, or that of bimodules).