(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.

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.

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?

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:

Have anyone ever introduced

Is there any introductory textbook (or lecture) on category theory that introduces natural transformation in this "homotopical" way rather then the classical onein any reference like a textbook or some lecture notes?

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:

Have anyone ever introduced natural transformation in this "homotopical" way rather then the classical one in any reference like a textbook or some lecture notes?

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)?