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!