I don't understand the need for defmulti.

  (defmethod my-add ((x integer) (y integer))
    (+ x y))

  (defmethod my-add ((x string) (y string))
    (concatenate 'string x y))

  CL-USER(8): (my-add 2 3)
  5
  CL-USER(9): (my-add "foo" "bar")
  "foobar"

Keep in mind you're dispatching on both arguments; the MY-ADD generic function is created for you and it's lurking in memory. You can add :before, :after and :around methods, which will allow you to get a hold of both arguments BEFORE the dispatch method is applied and after in a serial fashion, or AROUND the method dispatch altogether. This allows for stuff like logging, debugging, tracing, method wrapping and augmentation for moving-target type protocols and designs, etc.