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'\$$.