Let $F : \mathcal{C} \to \mathbf{Cat}$ be a lax 2-functor. Then we can form a category $\int F $ which is the Grothendieck construction on F. There's a number of resources detailing this construction, but none mentioning if we can get a 2-category, rather than a category this way. It seems like the natural construction would be to generate a 2-category - is this true? I'm looking for a reference.

Namely - it seems like the 2-cells in $\mathcal{C}$ in standard definitions don't really play any role at all in $\int F$, but it seems like they should correspond to 2-cells in $\int F$.

The closest constructions I found were in (https://arxiv.org/abs/2002.06055, Definition 10.7.2.) . They provide a bicategorical Grothendieck construction for a functor $F : \mathcal{C}^{op} \to \mathbf{Bicat^{ps}}$ - but of a rather strange kind. It does not use 2-cells in $\mathcal{C}$ and it also assumes certain equality of 1-cells in $\mathcal{C}$.

Another potential candidate is https://www2.irb.hr/korisnici/ibakovic/sgc.pdf, but it seems to be on a higher level of abstraction than I'm comfortable with. It also seems to talk about the Grothendieck construction of functors whose codomain is $\mathbf{2-Cat}$, rather than into $\mathbf{Cat}$. It seems like this is extra structure that is not needed for what I'm asking.

So, in short, if there is a lax functor $F : \mathcal{C} \to \mathbf{Cat}$, is there a way to make the Grothendieck construction into a 2-category? If so - is there reference with an explicit construction, showing in details what all the 2-cells would look like?