Definition:Whiskering

Definition
Let $B, C, D, E$ be categories.

Let $F, G : C \to D$ be covariant functors.

Let $\eta : F \to G$ be a natural transformation.

Natural Transformation followed by Functor
Let $H : D \to E$ be a covariant functor.

The right whiskering of $H$ and $\eta$ is the natural transformation $H \eta : H \circ F \to H \circ G$ between compositions of functors defined by $\paren {H \eta}_A = \map H {\eta_A}$ for $A \in C$.

Functor followed by Natural Transformation
Let $K : B \to C$ be a covariant functor.

The left whiskering of $\eta$ and $K$ is the natural transformation $\eta K : F \circ K \to G \circ K$ between compositions of functors defined by $\paren {\eta K}_A = \eta_{\map K A}$ for $A \in B$.

Also see

 * Definition:Whiskering Bifunctor
 * Definition:Whiskering Trifunctor