definition. cone [kostecki2011introduction, 4.9] [tt-0027]

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Let \({\cal C}^{{\cal J}}\) be a functor category, where \({\cal J}\) is a small category.

Let \(\Delta _O\) be a constant functor, which assigns the same object \(O\) in \({\cal C}\) to any object \(J\) in \({\cal J}\).

Let \(K \in \operatorname {Ob}({\cal J})\) and let \(j \in \operatorname {Arr}({\cal J})\) such that \(j: J \to K\). Let \(\mathscr {F}\) be any functor in \({\cal C}^{{\cal J}}\), i.e. it's a diagram in \({\cal C}\) of shape \({\cal J}\).

A natural transformation \(\pi : \Delta _O \to \mathscr {F}\), defined as a family of arrows \(\pi _J: O \to \mathscr {F}(J)\), such that the diagram commutes, is called a cone on the functor (diagram) \(\mathscr {F}\) with vertex \(O\).