We consider MacKay's renormalization operator
for pairs of area-preserving maps,
near the fixed point obtained in .
Of particular interest is the restriction R0 of this operator
to pairs that commute and have a zero Calabi invariant.
We prove that a suitable extension of R03
is hyperbolic at the fixed point, with a single expanding direction.
The pairs in this direction are presumably commuting,
but we currently have no proof for this.
Our analysis yields rigorous bounds on various "universal" quantities,
including the expanding eigenvalue.