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(Edit:) Edit2:) Some days ago I've read a post in nlab about $k$-transfor. In particular I belive have been interested by the discussion in the said post, because it seems to prove that "The power the homotopical definition of a question is all natural transformation should be the ideas right one (or other things you find out trying to answering it", this question gave to me at least a lot slight modification of ideas, I think I'll try to obtain just one lastit). After the various answers and comments I belive it's know On the time to make other end this last question:

Would it 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 good idea presenting new question:

Does anyone know the concept logical process that took Mac Lane and Eilenberg to give their (classical) definition of natural transformation?

Here I'm interested in the homotopical way topological/algebraic motivation that move those great mathematicians to such definition rather then the classical other onein a introductory textbook to category theory?

I'd like to see pros and cons.

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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?

(Edit:) I belive that "The power of a question is all the ideas or other things you find out trying to answering it", this question gave to me a lot of ideas, I think I'll try to obtain just one last. After the various answers and comments I belive it's know the time to make this last question:

Would it be a good idea presenting the concept of natural transformation in the homotopical way rather then the classical one in a introductory textbook to category theory?

I'd like to see pros and cons.

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