ct.category theory - How to characterize good "models" of a category

Let ${bf Cat}$ denote the category of small categories. Recall that for a category $mathcal{C}$ and a functor $Fcolonmathcal{C}to{bf Cat}$, the Grothendieck construction of $F$, which I'll denote $int F$, is a category and it comes equipped with a natural fibration $int Ftomathcal{C}$.

[For reference: The objects of $int F$ are pairs $(c,x)$ where $cin{bf Ob}(mathcal{C})$ and $xin{bf Ob}(F(mathcal{C}))$, and a morphism $(c,x)to(c',x')$ is a pair $(f,g)$ where $fcolon cto c'$ in $mathcal{C}$ and $gcolon F(f)(x)to x'$ in the category $F(c')$.]

Now, given a category ${mathcal D}$, I'll define a model of ${mathcal D}$ to be a pair $({mathcal C},F,e)$ where $mathcal{C}$ is a category, $Fcolon mathcal{C}to{bf Cat}$ is a functor, and $ecolonint Ftomathcal{D}$ is a natural isomorphism. I will sometimes leave out $e$ if it is obvious. Allow me to leave morphisms of models undefined, as I'm not sure what I want here. [Supplying an appropriate definition of morphisms between models of $mathcal{D}$ should be part of giving a good answer to this overflow question.]

Every category $mathcal{D}$ has two canonical models which I'll denote by $(mathcal{D},{ast})$ and $({ast},mathcal{D})$. The first is the functor $mathcal{D}to{bf Cat}$ that sends every object of $mathcal{C}$ to the terminal set, and the second is the functor $*to{bf Cat}$ that sends the terminal category to $mathcal{D}$.

But there may be many models $Fcolonmathcal{C}to{bf Cat}$ of $mathcal{D}$ that lie "in between" these two extreme cases, and some are "better than others" in the sense that the fibers of $mathcal{D}=int Ftomathcal{C}$ are non-trivial yet "comprehensible" in some human sense.

Question: What can you say about the (yet-undefined) category of models of $mathcal{D}$ that will clarify the above ideas?