# Definition:Cone (Category Theory)

## Definition

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}\[email protected][email protected]+2px{ C \ar[d]_*+{c_i} \ar[dr]^*+{c_j} \\ D_i \ar[r]_*+{D_\alpha} & D_j }\end{xy}$

is a commutative diagram.