  A second-quantized version of Mathieu moonshine leads to product formulae for
functions that are potentially genus-two Siegel Modular Forms analogous to the
Igusa Cusp Form. The modularity of these functions do not follow in an obvious
manner. For some conjugacy classes, but not all, they match known modular
forms. In this paper, we express the product formulae for all conjugacy classes
of $M_{24}$ in terms of products of standard modular forms. This provides a new
proof of their modularity.
