Definition:Category of Cones

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Let $\mathbf C$ be a metacategory.

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

The category of cones to $D$, denoted $\mathbf{Cone} \left({D}\right)$, is defined as follows:

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

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