I understand that functors map categories. But I'm confused by the notation, which mostly comes from the fact that categories contain both objects and morphisms.
Normally when I say $f \colon \mathbb{N} \to \mathbb{N}$, I would expect it to have exactly one mapping, e.g. $f \colon x \mapsto 2x$, but for functors, I have two mappings:
$$ \begin{align} & F \colon \mathcal{C} \to \mathcal{D} \\ & F \colon X \mapsto F(X) \\ & F \colon (f \colon X \to Y ) \mapsto (F(f) \colon F(X) \rightarrow F(Y)) \end{align} $$
Is this correct? If I have to map over all elements of $\mathcal{C}$, should this also include the composition $\circ$ ?
yes, that's correct. And that's include the composition. if we have g:Z→X F:(f∘g:Z→Y)↦(F(f∘g):F(Z)→F(Y)) and that's called covariant Functor, and one example of it, is homology functor. furthermore, there's other type called contravariant functor as the following F:C→D F:X↦F(X) F:(f:X→Y)↦(F(f):F(Y)→F(X)).