# Composition of a natural transformations with a functor--whiskering [duplicate]

If $$M$$ is an endofunctor on a category $$\cal K$$ and $$\eta:Id_{\cal K}\to M$$ is a natural transorfmation, what is the difference between $$\eta M$$ and $$M\eta$$, and how these two (componentwise) are defined on an object $$A$$ in $$\cal K$$ ?

$$(\eta M)_A=\eta_{M(A)}$$ and $$(M\eta)_A=M(\eta_A)$$
• Great. Will somethig change in your formulas if the n.t. $\eta$ were not from an Identity, but from a general endofunctor $F$ on $\cal K$ ? – user175304 Jul 4 '19 at 14:26
• Not essentially. It is in general that the component of $\sigma G$ and $G\sigma$ on object $A$ are $\sigma_{G(A)}$ and $G(\sigma_A)$ respectively (where $\sigma$ denotes a natural transformation and $G$ a functor. It is by definition. – drhab Jul 4 '19 at 14:31