(Edit #1 after Carlo's response)
It is often claimed that the notion of natural transformations existed in mathematical vocabulary long before it had a definition. In fact, I quoted the statement in italic from [1, p. 2]. As another example, in [2, p. 70] Ralf Kromer says: The claim is that there was, at the time when [3] was written, a current informal parlance consisting in calling certain transformations natural and that Mac Lane and Eilenberg tried (and succeeded) to grasp this informal parlance mathematically.
Three years after [3], Eilenberg and Maclane discovered the fact that this notion could be mathematically defined in [4].
However, Ralf Kromer casts doubt on the above mentioned claim, due to lack of evidence (see: [2, p. 70]).
(Edit #2 after Eric's comment) My question is, can you supply an evidence of use of the phrase "natural transformations" or its variants in mathematical literature prior to [3]? I must add that in [2] Kromer gives a number of examples of use of phrases such as "natural homomorphism" or "natural projection" in the literature prior to or around the same time as [3], but in each case they turn out to have different meanings. So I am asking for example(s) of use of the phrase "natural transformations", which are really natural transformations.
References:
- Peter Freyd: Abelian Categories (1964).
- Ralf Kromer: Tool and Object: a history and philosophy of category theory (2007).
- Samuel Eilenberg and Saunders Maclane: Group extensions and homology, Annals Math. (2) 43, p.p. 757–831 (1942).
- Samuel Eilenberg and Saunders Maclane: General theory of natural transformations, Trans. AMS, 58, p.p.: 231-294 (1945).