Definition:Morphism of Cones

Definition

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 if and only if, for all objects $j$ of $\mathbf J$:

$\begin{xy}\[email protected][email protected]+2px{ C \ar[r]^*+{f} \ar[dr]_*+{c_j} & C' \ar[d]^*+{c'_j} \\ & D_j }\end{xy}$

is a commutative diagram.