In the second half of the section "Operations with Natural Transformations" of the wikipedia article on natural transformations, they define the operation taking a natural tranformation $\eta:F\to G$ in the functor category $Cat(C,D)$ and functor $H:D\to E$ which produces a new natural transformation $(H\circ F)\to (H\circ G)$ in $Cat(C,E)$ by applying the functor to every coordinate of the original natural transformation. This operation frequently comes up in the definition of an adjunction.

Does this operation have a name? It's not the "Godement multiplication", but might be related to it.

Is there an agreed-upon symbol for this operation, other than juxtaposition?

*(Minor rant: I think that using juxtaposition for this operation is a horribly cruel thing to do to people learning category theory. Juxtaposition is already used for composition of morphisms -- and therefore for composition of functors-with-functors and naturaltransformations-with-naturaltransformations, since they too are the morphisms in $Cat$ and functor categories, respectively. Recycling juxtaposition yet again for this (non-associative!) operation on functors and natural transformations is just asking for trouble.)*

isassociative. $\endgroup$b)*c=a*(bc) is a well-sorted equation. However, in this case, the arguments are of different sorts (one is a functor, one is a natural transformation). What extension of the usual meaning of associativity do you have in mind here? $\endgroup$bv it doesn't matter whether I interpret that as two scalar multiplications or scalar multiplication by a product. $\endgroup$1more comment