\begin{definition}[Definition:Morphism of Cones]
Let $\mathbf C$ be a metacategory.
Let $D: \mathbf J \to \mathbf C$ be a $\mathbf J$-diagram in $\mathbf C$.
Let $\struct {C, c_j}$ and $\struct {C', c'_j}$ be cones to $D$.
Let $f: C \to C'$ be a morphism of $\mathbf C$.
Then $f$ is a '''morphism of cones''' {{iff}}, for all objects $j$ of $\mathbf J$:
::$\begin{xy}\xymatrix@+0.5em@L+2px{
 C
  \ar[r]^*+{f}
  \ar[dr]_*+{c_j}
&
 C'
  \ar[d]^*+{c'_j}
\\ &
 D_j
}\end{xy}$
is a commutative diagram.
\end{definition}