  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.
