  Using a realistic multi-band model for two holes doped into a CuO$_2$ layer,
we devise a method to turn off the magnon-mediated interaction between the
holes. This allows us to verify that this interaction is attractive, and
therefore could indeed be (part of) the superconducting glue. We derive its
analytical expression and show that it consists of a novel kind of
pair-hopping+spin-exchange terms. Its coupling constant is fitted from the
ground state energy obtained with variational exact diagonalization, and it
faithfully reproduces the effect of the magnon-mediated attraction in the
entire Brillouin zone. For realistic parameter values, this effective
interaction is borderline strong enough to bind the holes into preformed pairs.
