2-natural transformation?
homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
In a 2-category (and more generally in higher category theory) 2-morphisms have a composition along the 1-morphisms that they go between.
This is in contrast to the other composition operation, along objects, which is called horizontal composition.
In the 2-category Cat, 2-morphisms are natural transformations and vertical composition is the composition of these, the composition in the corresponding functor category.
For an abelian group and its double delooping 2-groupoid, vertical composition (as well as horizontal composition) is given by the group product operation in .