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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.)

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  • $\begingroup$ This operation is associative. $\endgroup$ – Toby Bartels Aug 22 '11 at 17:49
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    $\begingroup$ @Toby, associativity is generally considered to be a property of binary operations whose arguments are of the same sort, so (ab)*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? $\endgroup$ – Adam Aug 22 '11 at 18:53
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    $\begingroup$ The definition of associativity I usually use in this case is "you can write a string of symbols without ambiguity". E.g. I might say that vector-space scalar multiplication is associative because if I see abv it doesn't matter whether I interpret that as two scalar multiplications or scalar multiplication by a product. $\endgroup$ – Ben Millwood May 21 '13 at 9:07
  • $\begingroup$ Isn't associativity just an algebra for the non-empty list monad? $\endgroup$ – Andrej Bauer Apr 26 '17 at 19:09
  • $\begingroup$ Adam, this is ancient history I realize, but Toby might have had in mind the associativity of the horizontal composition of three (horizontally composable) 2-cells where the two on the ends are identities $1_F$ , $1_H$ of 1-cells $F, H$. $\endgroup$ – Todd Trimble Apr 26 '17 at 19:35
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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. This terminology is pretty widespread in the categorical community.

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