Let $\mathbf{Cat}$ denote the (strict) $2$-category of categories, functors, and natural transformations. Let $F: C \to \mathbf{Cat}$ be a pseudofunctor from a $1$-category $C$ to $\mathbf{Cat}$. The associated Grothendieck construction is defined as the oplax colimit of $F$. This is an object $\int F$ of $\mathbf{Cat}$, i.e. a category. This category can be described concretely as follows. Its objects are pairs $(c, a)$ where $c$ (resp. $a$) is an object of $C$ (resp. $F(c)$). Morphisms are given by pairs $(f: c \to c', \phi: F(f)(a) \to a')$.
I would like to replace $\mathbf{Cat}$ with a more general $2$-category and retain a description of $\int F$ similar to the element-wise one above.
The Grothendieck construction is an oplax colimit, so we could try to replace $\mathbf{Cat}$, which I believe has all ((op)lax) (co)limits, by a general $2$-category $\mathscr D$ which has all oplax colimits, or at least oplax colimits of shape $F$. In this case, the oplax colimit is not itself a category, hence does not admit a characterisation as given above in the case of $\mathbf{Cat}$. What conditions do we need to put on $\mathscr D$ to get the desired description?
There is a notion of a concrete category, i.e. a category $\mathcal D$ which admits a faithful functor to the category of sets. I assume one can define a concrete $2$-category as a $2$-category equipped with a $2$-functor to $\mathbf{Cat}$ which satisfies some properties -- I am not sure "faithful" is quite the correct term here. A functor $F: C \to \mathscr D$ into a concrete $2$-category which has oplax colimits of shape $F$ should then admit the same element-wise description of $\int F$.
I have a hunch that requiring $2$-concreteness of $\mathscr D$ would be too strong a condition: homotopy categories are not concretisable by a result of Freyd, so I assume that an appropriately defined homotopy $2$-category $\mathscr H$ is not $2$-concretisable either. Nevertheless, one could imagine a Grothendieck construction on $F: C \to \mathscr H$, assuming it exists, to have a characterisation in terms of objects and (homotopy classes of) maps.
There is a notion of generalised element in category theory. Could a generalisation of it be of use here, giving a weaker form of the desired characterisation of $\int F$? I don't know very much $2$-category theory and I apologise for the vagueness of my question. Any hints would be welcome.
