Smash Products for Non-cartesian Internal Prestacks

The smash product construction (or the Grothendieck construction) takes a functor (or prestack) $F \colon B^{op} \to \mathbf{Cat}$ and returns a fibration $p \colon A \to B$. In this paper, we develop an analogue of the smash product for prestacks internal to a non-cartesian monoidal category. Our construction simultaneously generalizes the Grothendieck construction for prestacks and smash products for $B$-module algebras over a bialgebra $B$. Further, taking fibers or coinvariants allows one to recover the original prestack.