\begin{definition}[Definition:Category of Cocones]
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:
{{DefineCategory
|ob = cocones to $D$
|mor = morphisms of cocones
|comp = Composition in $\mathbf C$
|id = $\operatorname{id}_{\left({C, c_j}\right)} := \operatorname{id}_C$, for a cocone $\left({C, c_j}\right)$
}}
\end{definition}