Lagrangian correspondences and the generalized Viterbo restriction functor

We study two kinds of functors of wrapped Fukaya categories: 1) the Viterbo restriction functor for an inclusion of a Liouville sub-domain; 2) the Lagrangian correspondence functor associated to the graph of the completion of the inclusion of the sub-domain, named the graph correspondence functor. We prove that these two functors agree on the sub-category where the Viterbo restriction functor is defined. We also extend the Viterbo restriction functor to the case of non-strongly-exact restrictions of Lagrangian submanifolds, which yields a natural object-wise deformation of the wrapped Fukaya category, constructed using another theory - linearized Legendrian homology. On the other hand, the graph correspondence functor is naturally defined on the whole wrapped Fukaya category, a priori taking values in a suitable enlargement of the wrapped Fukaya category having certain Lagrangian immersions as objects. We prove that the graph correspondence functor agrees with the extension of the Viterbo restriction functor.