Category theory - what's the relationship between these functors?

Question

I have a quite vague question: suppose to have categories that range over some collection of objects, like categories of sheaves over the collection of topological spaces, or the categories of vector spaces over the collection of fields; we can denote them as $\mathcal{C}(X)$, $\mathcal{D}(X)$ where $X\in \mathcal{Coll}$.
Suppose to find an equivalence $\mathcal{C}(X)\simeq \mathcal{D}(X)$, for any $X\in \mathcal{Coll}$, and to have for any $X,Y\in \mathcal{Coll}$, functors $F:\mathcal{C}(X)\rightarrow \mathcal{C}(Y)$, $G:\mathcal{D}(X)\rightarrow \mathcal{D}(Y)$, which are induced by the same $f:X\rightarrow Y$, for which the diagram $$ \require{AMScd} \begin{CD} \mathcal{C}(X) @>F>> \mathcal{C}(Y) \\ @VV{\simeq}V @VV{\simeq}V \\ \mathcal{D}(X) @>G>> \mathcal{D}(Y) \end{CD} $$ commutes.
Furthermore, if I create the category $\mathcal{C}$, whose objects are all $\mathcal{C}(X)$, for $X\in \mathcal{Coll}$, and morphisms the functors $F:\mathcal{C}(X)\rightarrow \mathcal{C}(Y)$ as above, and I repeat the same procedure in order to create the category $\mathcal{D}$, I can show that there is an equivalence $C\simeq D$.
What I fail to understand is the accurate relation between the functors $F,G$.
Can someone help me?

Answer 1

Let $A$ be a category, and consider functors $F, G : A^{\text{op}} \to \mathbf{Cat}$, i.e. indexed categories. Then $F$ and $G$ are equivalent as indexed categories if there is a 2-natural equivalence of $F$ and $G$, which is approximately the condition you describe (though, more generally, we can consider pseudonatural equivalences). For any such functor, we can construct a category, together with a fibration over $A$, and these categories will be equivalent if the indexed categories are. This is the content of the Grothendieck construction.