A cone \((C,c_*)\) over a diagram \(\mathcal{J}\xrightarrow{D}\mathcal{C}\) and the category \(\mathbf{Cone}(D)\)
We require:
An object \(C \in Ob(\mathcal{C})\)
For each object \(j \in \mathcal{J}\), a morphism \(C \xrightarrow{c_j}D(j)\).
The following property must be satisfied:
\(\forall f \in \mathcal{J}(j,k):\) \(c_k=c_j;D(f)\)
A morphism of cones is a morphism \(C \xrightarrow{a} C'\) in \(\mathcal{C}\) such that, for all \(j \in \mathcal{J}\), we have \(c_j=a;c'_j\).
Cones and their morphisms form a category.