Let $C$ and $D$ be categories, and let $F,G:C\rightarrow D$ be functors. Then a natural transformation $\tau$ from $F$ to $G$ is a family of morphisms $\{\tau_x\}_{x\in C}$ where for each $x\in X$, $\tau_x$ is a morphism between $F(x)$ and $G(x)$. And this family has to satisfy a certain commutative diagram. But I'm wondering whether natural transformations can be viewed in a different way.
An image of a functor isn't always a subcategory, but suppose that $F(C)$ and $G(C)$ are subcategories of $D$. Then my question is, can $\tau$ be viewed as a functor between the categories $F(C)$ and $G(C)$? It sends $F(x)$ to $G(x)$ for any object $x\in C$, and it sends $F(f)$ to $\tau_y\circ F(f)$ for any morphism $f:x\rightarrow y$ in $C$.
Or does none of this make sense?
The functor you have proposed is not necessarily well-defined; consider when there exist $x,x' \in C$ with $Fx=Fx'$ but $Gx \neq Gx'$.
I encourage you to research the notion of comma category. Very briefly, given functors $F: \mathscr{A} \rightarrow \mathscr{C}$ and $G:\mathscr{B} \rightarrow \mathscr{C}$, the objects of the comma category $(F \Rightarrow G)$ are morphisms $(A,B,f:FA \to GB)$ in $\mathscr{C}$ and the morphisms of the comma category $(F \Rightarrow G)$ are pairs $(j:A \rightarrow A',k:B \rightarrow B') \in \mathscr{A}(A,A') \times \mathscr{B}(B,B')$ such that the obvious square commutes. Notice that the objects of $(F \Rightarrow G)$ keep track of the objects $A \in \mathscr{A}, B \in \mathscr{B}$, and do not work only with the objects in the image, and that an arbitrary $f:FA \rightarrow GB$ need not be in the image of $F$.
Suppose $F,G:\mathscr{A} \rightarrow \mathscr{B}$. Let $\mathfrak{N}$ denote the collection of natural transformations $F \rightarrow G$. Given a natural transformation $\alpha: F \rightarrow G$, we can form the functor $T_{\alpha}:A \rightarrow (F \Rightarrow G)$ with $TA=(A,A,\alpha_A)$ and $Tf=(f,f)$. Let $\mathfrak{R}$ denote the collection of functors $T:C \rightarrow (F \Rightarrow G)$ with the property that $\forall A \in \mathscr{A}$ $TA=(A,A,f_A)$ for some $f_A:FA \to GA$ and $\forall h \in \mathscr{A}(A,A') \ \ Th=(h,h)$. Given such a functor $T$, we can form the collection of morphisms $\alpha_T = \left( (\alpha_T)_A \right)_{A \in \mathscr{A}} = \left( f_A \right) _{A \in \mathscr{A}}$.
https://ncatlab.org/nlab/show/comma+category#examples asserts that this $\alpha_T$ is a natural transformation, and therefore that this is a one-to-one correspondence between $\mathfrak{N}$ and $\mathfrak{R}$. I am not sure about this as the naturality condition for $\alpha_T$ seems to be identical to stating that $\forall A,A' \in \mathscr{A}$ $(F \Rightarrow G)(TA,TA') = \mathscr{A}(A,A') \times \mathscr{A}(A,A')$, which is not necessarily the case.