TITLE: Functor of points and category theory
QUESTION [18 upvotes]: I am trying to read the section on Functor of points from Eisenbud - Harris (and I also referred to Mumford's book). They all motivate functor of points this way :
In general, for any object $Z$ of a category $\mathcal{X}$, the association $X\mapsto\textrm{Hom}_\mathcal{X}(Z,X)$ defines a functor $\phi$ from the category $\mathcal{X}$ to the category of sets. (We wish to identify $\textrm{Hom}_\mathcal{X}(Z,X)$ with the point set $X$).  
But the book says that it is not satisfactory to call the set $\textrm{Hom}_\mathcal{X}(Z,X)$, the set of points of $X$ unless this functor $\phi$ is faithful.
I don't understand this statement. If $\phi$ is not faithful, they have given an example in the case of category of $CW$-complexes, where $\textrm{Hom}_\mathcal{X}(Z,X)$ cannot be identified with $X$. But I am not able to understand why $\textrm{Hom}_\mathcal{X}(Z,X)$ can b identified with $X$ if $\phi$ is faithful.
If $\phi$ is faithful, then there is an injection from $\textrm{Hom}_\mathcal{X}(Z,X)\longrightarrow\textrm{Hom}_{Sets}(\textrm{Hom}_\mathcal{X}(Z,Z), \textrm{Hom}_\mathcal{X}(Z,X))$. But what does this tell us? Any help will be appreciated!

REPLY [5 votes]: I may as well make this an answer; a functor $\hom_{\mathcal{X}}(Z,-)$ is faithful iff $Z$ is a generator of $\mathcal{X}$. Saying that you can "identify" an object $A$ with $\hom(Z,A)$ is perhaps a bit of an abuse of language; the idea is that the morphisms $Z\to A$ tell you everything about $A$ in $\mathcal{X}$. It also gives you an "underlying functor" that turns your category into a concrete category over $\mathbf{Set}$, so you can identify $\hom(Z,A)$ with $A$ in roughly the same sense you can identify a group with $\hom_{\mathbf{Group}}(\mathbb{Z},G)$. It gives you a way to view the abstract objects of the category as structured sets, and gives a nice canonical choice of set.
I'm not familiar with CW complexes, so I can't say whether there's some consideration more specific to that category. But I think these are general considerations for why you might think of an object as identifiable with its image under a faithful hom hom functor. Hopefully this made sense, let me know if I should expand on anything.