nLab
whiskering

Whiskering

Idea

In a 2-category, the horizontal composition of a 2-morphism with 1-morphisms is sometimes called whiskering.

Whiskering from the left with an equivalence and from the right with an inverse equivalence is a conjugation action of equivalences on 2-morphisms.

Examples

An important use of whiskering is the usual definition of adjoint functors via the triangle identities: in Cat whiskering is the composition of a functor with a natural transformation to produce a natural transformation.

If we identify a functor or 1-morphism with its identity natural transformation or identity 2-morphism?, then whiskering is a special case of horizontal composition, and composition of 1-morphisms is a special case of whiskering.

In detail:

References