Let \({\cal C}\) be a category, \({\cal J}\) a small category, and \(\mathscr {D} : {\cal J} \to {\cal C}\) a diagram in \({\cal C}\) of shape \({\cal J}\).
A cone on \(\mathscr {D}\) is an object \(V \in {\cal C}\) (the vertex of the cone) together with a family
\[
\left (V \xrightarrow {\pi _J} \mathscr {D}(J)\right )_{J \in {\cal J}}
\]
of arrows in \({\cal C}\) such that for all arrows \(J \to J'\) in \({\cal J}\), the diagram
commutes.
The family of arrows are components of a natural transformation \(\pi : \Delta _V \to \mathscr {D}\), i.e. from the constant functor ( which assigns the same object \(V\) to any object \(J_i\) in \({\cal J}\)) to diagram functor \(\mathscr {D}\).
For simplicity, we refer to a cone by "a cone \((V, \pi )\) on \(\mathscr {D}\)".
the natural transformation \(\alpha : \mathscr {F} \to \mathscr {G}\), denoted
is a collection of arrows \(\left \{\alpha _X : \mathscr {F}(X) \to \mathscr {G}(X) \right \}_{X \in \operatorname {Ob}({\cal C})}\) in \({\cal D}\) which satisfies naturality, i.e. makes the diagram
commute for every arrow \(f : X \to X'\) in \({\cal C}\). The arrows \(\left \{\alpha _X\right \}_{X \in \operatorname {Ob}({\cal C})}\) are called the components of the natural transformation.