Let be a diagram in a category .
If is an object of , a cone from to is a natural transformation
where denotes the constant functor.
In other words, a cone consists of morphisms (called the components of the cone)
one for each object of , which are compatible with all the morphisms of the diagram, in the sense that each diagram
commutes.
It’s called a cone because one pictures as sitting at the vertex, and the diagram itself as forming the base of the cone.
A cocone in is precisely a cone in the opposite category .