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.
