= Solution
Take the transition temperature $T_i\sim\eta$ and <radiation-dominated universe>. With $g_*\simeq100$ relativistic <degrees of freedom>, $\rho_{r,i}=(\pi^2/30)g_*T_i^4$ and the <Friedmann equation> gives
$$
H_i\simeq1.66\sqrt{g_*}\frac{\eta^2}{m_{\rm pl}},\qquad
\boxed{t_i\simeq\frac{0.301m_{\rm pl}}{\sqrt{g_*}\eta^2}
\simeq2.4\times10^{-39}\,\mathrm{s}\left(\frac{10^{16}\,\mathrm{GeV}}\eta\right)^2.}
$$
A saturated causal correlation length is of order the radiation-era horizon $H_i^{-1}=2t_i$. Write $C_n\sim1$ for the geometric/formation efficiency and $C_M=M_M/\eta$. One monopole per horizon volume gives the <horizon-density monopole abundance bound> estimate
$$
\boxed{n_i\sim C_nH_i^3=C_n(1.66)^3g_*^{3/2}\frac{\eta^6}{m_{\rm pl}^3},\qquad
\rho_i\sim M_Mn_i=C_nC_M(1.66)^3g_*^{3/2}\frac{\eta^7}{m_{\rm pl}^3}.}
$$
The first quantity is number density and the second is defect energy density. Relative to the initial radiation <critical density>,
$$
\frac{\rho_i}{\rho_{r,i}}\sim13.9\,C_nC_M\sqrt{g_*}\left(\frac\eta{m_{\rm pl}}\right)^3.
$$
This can initially be small even when the surviving monopole abundance is unacceptable today. Causality supplies an order-of-magnitude lower abundance estimate under the stipulated saturation; it is not a precise statistical count of defects.
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