definition. cone as a natural transformation [leinster2016basic, eq. 6.1] [tt-0040]

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Now, given a diagram \(\mathscr {D}: {\cal J} \to {\cal C}\) and an object \(V \in {\cal C}\), a cone on \(\mathscr {D}\) with vertex \(V\) is simply a natural transformation from the diagonal functor \(\Delta _V\) to the diagram \(\mathscr {D}\).

Writing \(\operatorname {Cone}(V, \mathscr {D})\) for the set of cones on \(\mathscr {D}\) with vertex \(V\), we therefore have \[ \operatorname {Cone}(V, \mathscr {D})=[{\cal J}, {\cal C}] (\Delta _V, \mathscr {D}) . \]

Thus, \(\operatorname {Cone}(V, \mathscr {D})\) is functorial in \(V\) (contravariantly) and \(\mathscr {D}\) (covariantly).