# Does the Grothendieck construction satisfy Fubini's thorem

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

Another good source on these topics is Emily Riehl's book. The Fubini result is extended to the setting of $$n$$-categories by this thesis.