lemma. limit via representation [leinster2016basic, 6.1.1] [tt-003Y]

✍️source

Let \({\cal J}\) be a small category, \({\cal C}\) a category, and \(\mathscr {D}: {\cal J} \to {\cal C}\) a diagram. Then there is a one-to-one correspondence between

• limit cones on \(\mathscr {D}\)
• representations of the natural transformation Cone

Briefly put: a limit \((V, \pi )\) of \(\mathscr {D}\) is a representation of \([{\cal J}, {\cal C}] (\Delta _{-}, \mathscr {D})\).

Diagramatically,

It implies that \[\operatorname {Cone}(\mathrm {-}, \mathscr {D}) \cong {\cal C}\left (\mathrm {-}, \lim \limits _{\leftarrow {\cal J}} \mathscr {D} \right )\] for any \(\mathrm {-} \in {\cal C}\).