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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 $\cal K$ ?

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$(\eta M)_A=\eta_{M(A)}$ and $(M\eta)_A=M(\eta_A)$

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  • $\begingroup$ Great. Will somethig change in your formulas if the n.t. $\eta$ were not from an Identity, but from a general endofunctor $F$ on $\cal K$ ? $\endgroup$ – user175304 Jul 4 '19 at 14:26
  • $\begingroup$ 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. $\endgroup$ – drhab Jul 4 '19 at 14:31
  • $\begingroup$ Could you please have a look at this question ? $\endgroup$ – user175304 Jul 5 '19 at 14:21