I am reading [this paper](http://www.ams.org/journals/proc/1972-036-02/S0002-9939-1972-0334212-5/S0002-9939-1972-0334212-5.pdf) on Homotopy for functors by Ming-Jung 
 Lee.


The author gives a definition (at the beginning of section $3$) as follows : 



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> Let $\varphi,\varphi':\Lambda\rightarrow \Gamma$ be covariant functors of small categories. We say that $\varphi$ is homotopic to $\varphi'$ if there are covariant functors $\varphi\_i:\Lambda\rightarrow \Gamma$ for $i=0,1,\cdots,n$ such that $\varphi\_0=\varphi,\varphi\_n=\varphi'$ and for each $i$, there is a natural transformation between $\varphi\_i$ and $\varphi\_{i+1}$.
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Given two natural transformations, there is an obvious way to compose them which gives a natural transformation.



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> What is the point of considering the finite collection of natural transformations between given two functors. We can just define that two functors are homotopic if there is a natural transformation between them. 
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I do not understand what I am missing. Any suggestions are helpful.



Homotopy for functors
The author means there is a zigzag of natural transformations. That is, "a natural transformation between $\varphi\_i$ and $\varphi\_{i+1}$" is intended to be nonspecific as to the direction of the transformation: it could go from $\varphi\_i$ to $\varphi\_{i+1}$ or from $\varphi\_{i+1}$ to $\varphi\_{i}$.


This is a reasonable notion of "homotopy" between functors because upon passing to geometric realizations / classifying spaces, any natural transformation induces a homotopy in the topological sense, and homotopies in the latter sense can always be reversed as well as composed; thus any zigzag of natural transformations between functors induces a single homotopy between their geometric realizations.


