Suppose we are given a functor 


$F:(A\times B)^{\operatorname{op}}\to \operatorname{Set}$.


It's well-known that the Grothendieck construction in this case evaluates as 


$\int\_{A\times B}F = (A\times B)/F$.


We could also apply this construction pointwise to obtain a functor


$\int\_A F:B^{op}\to \operatorname{Cat}$


sending $b\mapsto A/F(b)$


and similarly


$\int\_B F:A^{op}\to \operatorname{Cat}$


We can apply the Grothendieck construction again to each of these functors to obtain categories


$\int\_A\int\_B F$


and


$\int\_B\int\_A F$


Is it the case that $\int\_A \int\_B F\cong \int\_B \int\_A F\cong \int\_{A\times B} F$?



Does the Grothendieck construction satisfy Fubini's thorem
Yes. See section 4.2 of [this](https://www.math.univ-paris13.fr/~harpaz/grothendieck.pdf) paper by Harpaz and Prasma, which also extends the result to the setting of model categories.


Another good source on these topics is [Emily Riehl's book](https://web.math.rochester.edu/people/faculty/doug/otherpapers/riehl.pdf). The Fubini result is extended to the setting of $n$-categories by [this](https://pdfs.semanticscholar.org/2ef5/c935ded8444717a43ca9e977fced8cee31ea.pdf) thesis.


