Matthew wrote:

> For a category \\(\mathcal{C}\\) an endofunctor \\(F: \mathbf{Hom}(\mathcal{C}) \to \mathbf{Hom}(\mathcal{C})\\) maps morphisms to morphisms.

A purely botational comment:

I guess you're using \\(\mathbf{Hom}(\mathcal{C})\\) to mean the set of all morphisms in \\(\mathcal{C}\\). The usual notation for the set of all morphisms in \\(\mathcal{C}\\) is \\(\mathbf{Mor}(\mathcal{C})\\), so I recommend this; it goes along with \\(\mathbf{Ob}(\mathcal{C})\\) for the set of objects.

I've never seen anyone write \\(\mathbf{Hom}(\mathcal{C})\\). We often see \\(\mathrm{Hom}(c,c')\\), which means the set of morphisms from some object \\(c\\) to some object \\(c'\\).