Definition:Morphism of Cocones

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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 cocones from $D$.

Let $f: C \to C'$ be a morphism of $\mathbf C$.

Then $f$ is a morphism of cones if and only if, for all objects $j$ of $\mathbf J$:

$\begin{xy}\xymatrix@+0.5em@L+3px{ D_j \ar[d]_*+{c_j} \ar[dr]^*+{c'_j} \\ C \ar[r]_*+{f} & C' }\end{xy}$

is a commutative diagram.