I am new to category theory but I think I have a basic understanding of functors.
Is there a way of proving the existence of, and perhaps fully characterizing, a functor $\mathcal{F}$ mapping a category $\mathcal{C}$ to $\mathcal{D}$. The categories $\mathcal{C}$ and $\mathcal{D}$ have undirected graphs as objects and homomorphisms as morphisms. The objects of each category are generated from an initial graph, for instance let us call it $G_0(\mathcal{C})$ for $\mathcal{C}$ and $G_0(\mathcal{D})$ for $\mathcal{D}$.
As an example of how the categories are constructed, let us take $G_0(\mathcal{C})=ACBE$ to be constructed from four nodes $\{A, B, C, E\}$ connected by edges $\{(A, B), (A, E), (B, C)\}$. The morphisms initially map $G_0(\mathcal{C})$ to new homomorphic graphs. In this case, we would get 16 homomorphisms, yielding 16 graphs:
$G_0(\mathcal{C}) \rightarrow AABB$
$G_0(\mathcal{C}) \rightarrow AABE$
$G_0(\mathcal{C}) \rightarrow ACBB$
$G_0(\mathcal{C}) \rightarrow ACBE$
$G_0(\mathcal{C}) \rightarrow AAEB$
$G_0(\mathcal{C}) \rightarrow AAEE$
$G_0(\mathcal{C}) \rightarrow CABB$
$G_0(\mathcal{C}) \rightarrow CCBB$
$G_0(\mathcal{C}) \rightarrow BBAA$
$G_0(\mathcal{C}) \rightarrow BBAC$
$G_0(\mathcal{C}) \rightarrow BEAA$
$G_0(\mathcal{C}) \rightarrow BEAC$
$G_0(\mathcal{C}) \rightarrow BBCA$
$G_0(\mathcal{C}) \rightarrow BBCC$
$G_0(\mathcal{C}) \rightarrow EBAA$
$G_0(\mathcal{C}) \rightarrow EEAA$
I thought about gluing the redundant nodes together, so that the first graph AABB is formed of nodes $\{A, B\}$, AABE is formed of nodes $\{A, B, E\}$, etc. Similarly, it is possible to continue applying homomorphisms on the second graphs (AABB, AABE, ACBB, etc.) and obtain new ones until all the graphs in one path collapse to one edge $G_0(\mathcal{C}) \rightarrow AABB \rightarrow AB$ (I am not sure if this is true). The whole process will result in a tree-like representation with $G_0(\mathcal{C})$ being the root node and the leafs being final edges (again I am not sure if this is true).
So, for $G_0(\mathcal{C})$ and $G_0(\mathcal{D})$ with $G_0(\mathcal{C}) \neq G_0(\mathcal{D})$, my goal is to compute (or derive?) functor(s) that can map any object or morphism of $\mathcal{C}$ into its corresponding one in $\mathcal{D}$. Of course, such functor(s) have to ensure composition: $\mathcal{F}(f \circ g) = \mathcal{F}f \circ \mathcal{F}g$ with $f,g \in Mor(\mathcal{C})$
Maybe it helps, if you think about functors as "functions between categories". If you know a bit of linear algebra, one example might be the dualizing functor $(-)^*$, which assigns each vector space its dual vector space.
\begin{align} (-)^*: \textbf{Vect}_k & \rightarrow \textbf{Vect}_k \\ V &\mapsto V^* = Hom_k(V, k) \\ (\varphi:V \rightarrow W) & \mapsto (\varphi^*: W^*\rightarrow V^*) \end{align}
$\varphi^*$ is defined by $\varphi^*(f) = f\circ \varphi$, i.e. for $v \in V$ and $f \in W^*$, $\varphi^*(f) (v) = f(\varphi(v))$.
So "calculating" the functor is the same as calculating the dual of a vector space, whenever this is necessary.