To import a map axes, it is only sufficient to change the parent of a map axes as follows:
% ... MapAxes is created in a new figure
% Change the Parent of the map axes
TempParent = MapAxes.Parent;
MapAxes.Parent = UIAxes.Parent;
% fit the size
MapAxes.Units = 'pixel';
MapAxes.Position = UIAxes.Position;
% delete the initial temperory UIAxes
delete(UIAxes);
% delete the temperory map figure
delete(TempParent);