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.)

  • $\begingroup$ This operation is associative. $\endgroup$ Aug 22, 2011 at 17:49
  • 1
    $\begingroup$ @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, 2011 at 18:53
  • 4
    $\begingroup$ 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. $\endgroup$ May 21, 2013 at 9:07
  • $\begingroup$ Isn't associativity just an algebra for the non-empty list monad? $\endgroup$ Apr 26, 2017 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, 2017 at 19:35

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.