Definition:Category of Cocones

Definition

Let $\mathbf C$ be a metacategory.

Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$.

The category of cocones from $D$, denoted $\mathbf{Cocone} \left({D}\right)$, is the category with:

 Objects: cocones to $D$ Morphisms: morphisms of cocones Composition: Composition in $\mathbf C$ Identity morphisms: $\operatorname{id}_{\left({C, c_j}\right)} := \operatorname{id}_C$, for a cocone $\left({C, c_j}\right)$