nLab
cone

Definition

Let $F:J\to C$ be a diagram in a category $C$.

If $c$ is an object of $C$, a cone from $c$ to $F$ is a natural transformation

$$T:\Delta (c)\to F$$T: \Delta(c) \to F

where $\Delta (c):J\to C$ denotes the constant functor.

In other words, a cone consists of morphisms (called the components of the cone)

$$T_j:c\to F(j),$$T_j: c \to F(j),

one for each object $j$ of $J$, which are compatible with all the morphisms $F(f):F(j)\to F(k)$ of the diagram, in the sense that each diagram

$$\begin{array}{ccccc}& & c& & \\ & T_j\swarrow & & \searrow T_k& \\ F(j)& & \stackrel{F(f)}{⟶}& & F(k)\\ \end{array}$$\array{{}&{}&c&{}&{} \\ {}& \mathllap{\scriptsize{T_j}}\swarrow &{}& \searrow\mathrlap{\scriptsize{T_k}} &{} \\ F(j) &{}&\stackrel{F(f)}{\longrightarrow} &{}& F(k) \\ }

commutes.

It’s called a cone because one pictures $c$ as sitting at the vertex, and the diagram itself as forming the base of the cone.

A cocone in $C$ is precisely a cone in the opposite category $C^{\mathrm{op}}$.