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!)
