What exactly is the naturalizer of a natural transformation Wikipedia says:

The naturalizer of η, nat(η), is the largest subcategory of C containing all the objects of C on which η restricts to a natural transformation.

I don't exactly understand what "on which $\eta$ restricts to" means. 
I want to illustrate my confusion with an example:
Given category 1, a discrete category with one object, called $X$; and a category $C$ with two objects with morphisms between them.
Now I say, I have two functors $F,G: 1 \to C$, and a natural transformation $\varepsilon: F \to G$.
$F(X) \neq G(X)$, that is they map to the two different objects of $C$. Now considering $\varepsilon_X: FX \to GX$, is 
$$nat(\varepsilon) = C$$
or
$$nat(\varepsilon) = \text{subcategory with } G(X), 1_{G(X)}$$ 
Because in the first case, the naturalizer is just the image of all the components of the natural transformation combined, while in the second case, it considers the subcategory you're restricted to, when you enter through the combination of the nat.trans. + first functor, instead of directly using the second functor.
 A: A natural transformation between functors $F,G\colon D\to C$ is a map from the objects of $D$ to the arrows of $C$. So $\text{nat}(\varepsilon)$ is a subbcategory of $D$, the domain category of your functors. Not of the codomain category $C$.
In your case, $\varepsilon$ is a map from objects of $1.$ Your domain category is $1.$ Your $\varepsilon$ is already a natural transformation, so the naturalizer is all of the domain, $\text{nat}(\varepsilon)=1.$
A: Given functors $F,G : C \to D$, the wikipedia article talks about an infranatural transformation $\eta$ from $F$ to $G$ to be a function from the objects of $C$ to the arrows of $D$ with the property that $F(X) \xrightarrow{\eta(X)} G(X)$.
If $C' \subseteq C$ is any subcategory, we can talk about the restriction of the functors $F$ (and $G$). $F|_{C'}$ is the functor $C' \to D$ we get simply by restricting the domain of $F$.
Put differently, if $i : C' \to C$ is the inclusion functor, then $F|_{C'} = F \circ i$.
Correspondingly, we also talk about the restriction of an infranatural transformation; $\eta|_{C'}$ is an infranatural transformation $F|_{C'} \to G|_{C'}$, and its defining function is the obvious one.
$\operatorname{nat}(\eta)$ maybe has a simpler definition: it is the subcategory of $C$ consisting of all of the objects of $C$ along with all of the arrows $f : X \to Y$ satisfying the identity $\eta(Y) F(f) = G(f) \eta(X)$.
