Generalisation of the Grothendieck construction for presheaves as a lax pullback It is well-known that for any presheaf $F \colon \mathcal{C}^{\mathrm{op}} \to \mathrm{Set}$, the category of elements (obtained by the so-called Grothendieck construction) of $F$ is a comma category $y/\ulcorner F \urcorner$ in the category $\mathrm{CAT}$ of 
$$\mathcal{C} \xrightarrow{y} \widehat{\mathcal{C}} \xleftarrow{\ulcorner F \urcorner} 1.$$
Question: does this generalise to presheaves $F \colon \mathcal{C}^{\mathrm{op}} \to \mathrm{Gpd}$ of groupoids (or even categories)?
Mere comma categories yield discrete fibrations, hence won't give the expected answer. So the question is whether some other construction with a similar flavour could do.
Probably, if the construction does generalise, then it will also work for pseudo-functors to $\mathrm{Gpd}$ or $\mathrm{Cat}$, but I'm really interested in strict functors.
Note: I'm half-aware of another universal property of the Grothendieck construction as an oplax colimit. Is it related? 
 A: Possibly the comma 2-category gives what you want.
If you think of $C^{op}\rightarrow Gpd$ as a functor of 2-categories (e.g. pass to the nerve for some model of $(\infty,2)$ categories) and then take the comma object 
$$
\matrix{(r\downarrow F)&\rightarrow& C \\
\downarrow & & \downarrow _{r} \\
1 & \rightarrow^F & Gpd^{C^{op}}}
$$
(where $r$ is the Yoneda imbedding.)
then an object is then a pair $(c\in C,\varphi:rc\rightarrow F)$, and a morphism is a pair 
$(\Psi:c\rightarrow c',M:\varphi\rightarrow \varphi'\circ r\Psi)$.
There is a canonical map from the Grothendiek construction into this comma category, and I suspect it is a homotopy equivalence, so that when you take the corresponding usual category you get the same thing.
A: Another way of phrasing the isomorphism of categories $\int^{\mathcal{C}}\mathcal{F}\congよ\downarrow[\mathcal{F}]$ is by saying that $\int^{\mathcal{C}}\mathcal{F}$ is the full subcategory of $\mathsf{PSh}(\mathcal{C})_{/\mathcal{F}}$ spanned by the representable presheaves. A quick way to prove this is as follows:

*

*First, note that $よ\downarrow[\mathcal{F}]$ is the category of elements of the functor $\mathsf{Nat}(h_{(-)},\mathcal{F})\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Sets}$.

*Then, the Yoneda lemma gives a natural isomorphism $\mathsf{Nat}(h_{(-)},\mathcal{F})\cong\mathcal{F}$. This is an isomorphism in $\mathsf{PSh}(\mathcal{C})$.

*As functors preserve isomorphisms, it follows that $\int^{\mathcal{C}}\colon\mathsf{PSh}(\mathcal{C}^{\mathsf{op}})\longrightarrow\mathsf{Cats}$ sends the isomorphism of presheaves $\mathcal{F}\cong\mathsf{Nat}(h_{(-)},\mathcal{F})$ to an isomorphism of categories $\int^{\mathcal{C}}\mathcal{F}\cong\underbrace{\int^{\mathcal{C}}\mathsf{Nat}(h_{(-)},\mathcal{F})}_{\congよ\downarrow[\mathcal{F}]}$.

To give such a description for the Grothendieck construction, we first pass to bicategories. There, given a pseudopresheaf $\mathcal{F}\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Cats}_{\mathsf{2}}$ from a bicategory $\mathcal{C}$ to the $2$-category $\mathsf{Cats}_{\mathsf{2}}$ of small categories, functors, and natural transformations, we can repeat the above strategy:

*

*This time, we consider the locally full sub-bicategory of $\mathsf{PseudoPSh}(\mathcal{C})_{/\mathcal{F}}$ spanned by the representable pseudopresheaves. This is the bicategory $よ(\mathcal{C})_{/\mathcal{F}}$ where




and where vertical and horizontal composition of $2$-morphisms is defined as in $\mathsf{PseudoPSh}(\mathcal{C})$ (we also have to specify the associators and unitors, but let's not since this description is already quite long).

*

*Now, note that $よ(\mathcal{C})_{/\mathcal{F}}$ is the bicategory of elements (as defined in arXiv:1212.6283) of the pseudopresheaf $\mathsf{PseudoNat}(\mathsf{h}_{(-)},\mathcal{F})\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Cats}_{\mathsf{2}}$.

*Just as functors preserve isomorphisms, functors of tricategories preserve biequivalences. We can then apply Proposition 3.3.6 of arXiv:1212.6283 to the equivalence $\mathsf{PseudoNat}(\mathsf{h}_(-),\mathcal{F})\cong\mathcal{F}$ provided by the bicategorical Yoneda lemma to obtain a biequivalence $\int^{\mathcal{C}}\mathcal{F}\cong\underbrace{\int^{\mathcal{C}}\mathsf{PseudoNat}(\mathsf{h}_{(-)},\mathcal{F})}_{\cong よ(\mathcal{C})_{/\mathcal{F}}}$.

Finally we connect the above back to the Grothendieck construction. To this end, observe that―just as the category of elements of a presheaf $\mathcal{F}\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Sets}$ agrees with the Grothendieck construction of $\mathcal{F}_{\mathsf{disc}}\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Cats}$―the Grothendieck construction of a pseudofunctor $\mathcal{F}\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Cats}_{\mathsf{2}}$ agrees with the bicategory of elements of $\mathcal{F}\colon\mathcal{C}^{\mathsf{op}}_\mathsf{bi}\longrightarrow\mathsf{Cats}_{\mathsf{2}}$, where $\mathcal{C}^{\mathsf{op}}_\mathsf{bi}$ is the discrete bicategory associated to $\mathcal{C}$ (this is $\mathcal{C}$ but with discrete $\mathsf{Hom}$-categories, i.e. for each $A,B\in\mathrm{Obj}(\mathcal{C})$, we have $\mathsf{Hom}_{\mathcal{C}^{\mathsf{op}}_\mathsf{bi}}(A,B)\overset{\mathrm{def}}{=}\mathrm{Hom}_{\mathcal{C}}(A,B)_{\mathsf{disc}}$).

All this is to say the following: the Grothendieck construction of a pseudofunctor $\mathcal{F}\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Cats}_{\mathsf{2}}$ is equivalent to the category $よ_\mathsf{disc}(\mathcal{C})_{/\mathcal{F}}$ where


(Sorry for the long answer!)
A: The answer seems to be "no": Consider a strong-/pseudofuntor $F:C^\mathrm{op}\to \mathrm{Cat}$.
The object "sets" are isomorphic: they are given by pairs $(x,a_x)$ with $x\in C$ and $a_x\in F(x)$. But arrows in the Grothendieck construction / category of elements are given by pairs
$$(f:x\to y,\varphi: a_x\to f^*a_y)$$
where $f^*:=F(f)$.
On the other hand: Arrows in the respective comma-category are given by triples
$$(f:x\to y, \Phi: F\to F, [\varphi]: \Phi_x(a_x) \to f^*a_y).$$
Composition is similar but the endo-transformation $\Phi$ does not need to be the identity.
These categories are in general not equivalent: Consider the case where $C=1$ for suitable $F$.
