MathGloss

A cone over a diagram F:JCF : \mathsf{J} \to \mathsf{C} with summit or apex cCc \in \mathsf{C} is a natural transformation λ:cF\lambda : c \Rightarrow F whose domain is the constant functor at cc. The components (λj:cFj)jJ(\lambda_j : c \to Fj)_{j\in \mathsf{J}} of the natural transformation are called the legs of the cone. Explicitly:

Dually, a cone under FF with nadir cc is a natural transformation λ:Fc\lambda : F \Rightarrow c, whose legs are the components $(\lambda_j : F_j \to c){j \in \mathsf{J}}.Thenaturalityconditionassertsthat,foreachmorphism. The naturality condition asserts that, for each morphism f : j \to kof of \mathsf{J},thetriangle, the triangle \xymatrix@=10pt{ Fj \ar[dr]{\lambda_j} \ar[rr]^{Ff} & & Fk\ar[dl]^{\lambda_k} \ & c}commutesin commutes in \mathsf{C}$.

SUGGESTION: cone under

From Category Theory in Context