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