A cone over a diagram F:J→C with summit c∈C is a natural transformation λ:Δ(c)⇒F from the constant functor Δ(c) to F.
Each component λj is called a leg.
The naturality square in this case looks like:
which is the same as
The triangular shape of this commutative diagram gives rise to the name “cone” for the object being discussed
Dually, a cone under F (or co-cones) with nadir c∈C is a natural transformation μ:F⇒Δ(c).
For any diagram F:J→C, we can create the cone functor
Cone(−,F):Cop→Set:c:(f:c→c′)↦Nat(Δ(c),F)=cones over F with summit c↦(μ↦η given by ηj=μjf)
Then the category of elements ∫Cone(−,F) gives rise to the category of cones over F whose objects are cones over F and whose morphisms are commutative diagrams