Functors of wrapped Fukaya categories from Lagrangian correspondences

We study wrapped Floer theory on product Liouville manifolds and prove that the wrapped Fukaya categories defined with respect to two different kinds of natural Hamiltonians and almost complex structures are equivalent. The implication is we can do quilted version of wrapped Floer theory, based on which we then construct functors between wrapped Fukaya categories of Liouville manifolds from certain classes of Lagrangian correspondences, by enlarging the wrapped Fukaya categories appropriately, allowing exact cylindrical Lagrangian immersions. For applications, we present a general K"{u}nneth formula, and also identify the Viterbo restriction functor with the functor associated to the completed graph of embedding of a Liouville sub-domain.