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 $\int_A \int_B F\cong \int_B \int_A F\cong \int_{A\times B} F$?