\begin{definition}[Definition:Cocone]
Let $\mathbf C$ be a metacategory.
Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$.
A '''cocone from $D$''' comprises an object $C$ of $\mathbf C$, and a morphism:
:$c_j: D_j \to C$
for each object of $\mathbf J$, such that for each morphism $\alpha: i \to j$ of $\mathbf J$:
::$\begin{xy}\xymatrix@+0.5em@L+2px{
 D_i
  \ar[r]^*+{D_\alpha}
  \ar[dr]_*+{c_i}
&
 D_j
  \ar[d]^*+{c_j}
\\ &
 C
}\end{xy}$
is a commutative diagram.
{{expand|when time comes, add def "cocone is natural trafo $D \to \kappa_C$ with $\kappa_C$ constant functor"}}
\end{definition}