nLab
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

For instance in Cat whiskering is the composition of a functor with a natural transformation to produce a natural transformation, If we identify a functor or morphism with its identity natural transformation or identity 2-morphism?, then whiskering is a special case of horizontal composition, and composition of morphisms is a special case of whiskering.

In detail:

• If $F,G:C\to D$ and $H:D\to E$ are functors and $\eta :F\to G$ is a natural transformation whose coordinate at any object $A$ of $C$ is ${\eta }_A$, then whiskering $H$ and $\eta$ yields the natural transformation $H\circ \eta :(H\circ F)\to (H\circ G)$ whose coordinate at $A$ is $H({\eta }_A)$.
• If $F:C\to D$ and $G,H:D\to E$ are functors and $\eta :G\to H$ is a natural transformation whose coordinate at $A$ is ${\eta }_A$, then whiskering $\eta$ and $F$ yields the natural transformation $\eta \circ F:(G\circ F)\to (H\circ F)$ whose coordinate at $A$ is ${\eta }_{F(A)}$.

References

• A MathOverflow question about whiskering