In the second half of the section "[Operations with Natural Transformations](http://en.wikipedia.org/wiki/Natural_transformation#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.)*



What is the name for the composition of a functor with a natural transformation?
It's called "whiskering" -- the 1-cells/functors composed on either side of the 2-cell/transformation look like "whiskers". See for example page 24 of [this paper](http://math.ucr.edu/home/baez/2rep.pdf). This terminology is pretty widespread in the categorical community.


