I'm reading Leinster's book and at the end of Section 1.3 he defines horizontal composition of natural transformations. I don't see any motivation for this definition, and the only application the author mentions is the construction of some functor $[\mathscr A',\mathscr A'']\times[\mathscr A,\mathscr A']\to[\mathscr A,\mathscr A'']$, which I'm not sure why I should care about.
On the other hand, it is clear why one needs the vertical composition -- at least to define composition in the functor category; and because of this I find the vertical composition quite an important notion.
My question is whether the horizontal composition is as important as the vertical one (e.g. is it used as frequently? If I skip this part of the text, will I be able to understand the basics of adjoints/representables/limits?) and, if it's possible to answer assuming only the material up to Section 1.3 of the aforementioned book, where/how can it be used?