Every text book I've ever read about Category Theory gives the definition of natural transformation as a collection of morphisms which make the well known diagrams commute.
There is another possible definition of natural transformation, which appears to be a categorification of homotopy:
given two functors $\mathcal F,\mathcal G \colon \mathcal C \to \mathcal D$ a natural transformation is a functor $\varphi \colon \mathcal C \times 2 \to \mathcal D$, where $2$ is the arrow category $0 \to 1$, such that $\varphi(-,0)=\mathcal F$ and $\varphi(-,1)=\mathcal G$.
My question is:
why doesn't anybody use this definition of natural transformation which seems to be more "natural" (at least for me)?
(Edit:) It seems that many people use this definition of natural transformation. This arises the following question:
Is there any introductory textbook (or lecture) on category theory that introduces natural transformation in this "homotopical" way rather then the classical one?
(Edit2:) Some days ago I've read a post in nlab about $k$-transfor. In particular I have been interested by the discussion in the said post, because it seems to prove that the homotopical definition of natural transformation should be the right one (or at least a slight modification of it). On the other end this definition have always seemed to be the most natural one, because historically category theory develop in the context of algebraic topology, so now I've a new question:
Does anyone know the logical process that took Mac Lane and Eilenberg to give their (classical) definition of natural transformation?
Here I'm interested in the topological/algebraic motivation that move those great mathematicians to such definition rather the other one.
Best Answer
The homotopy analogue definition of natural transformations has been known and used regularly since at least the late 1960's, by which time it was understood that the classifying space functor from (small) categories to spaces converts natural transformations to homotopies because it takes the category $I=2$ to the unit interval and preserves products. Composition of natural transformations $H\colon A\times I\to B$ and $J\colon B\times I\to C$ is just the obvious composite starting with $id\times \Delta: A\times I \to A\times I\times I$, just as in topology. (I've been teaching that for at least several decades, and I'm sure I'm not the only one.)