\begin{definition}[Definition:Colimit]
Let $\mathbf C$ be a metacategory.
Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$.
Let $\map {\mathbf {Cocone} } D$ be the category of cocones from $D$.
A '''colimit for $D$''' is an initial object in $\map {\mathbf {Cocone} } D$.
It is denoted by $\varinjlim_j D_j$; the associated morphisms $\iota_i: D_i \to \varinjlim_j D_j$ are usually left implicit.
\end{definition}