  Cheptea, Habiro and Massuyeau constructed the LMO functor, which is defined
on a certain category of cobordisms between two surfaces with at most one
boundary component. In this paper, we extend the LMO functor to the case of any
number of boundary components, and our functor reflects relations among the
parts corresponding to the genera and boundary components of surfaces. We also
discuss a relationship with finite-type invariants and Milnor invariants.
