The singular differential equations for finite temperature relativistic magnetohydrodynamic (MHD) winds integrate to two algebraic equations when the source magnetic field is a monopole. This simplification enables an extensive characterization of the asymptotic wind solutions in terms of source parameters. We will consider only the critical solutions-those that pass smoothly through both an intermediate (Alfvenic) and a fast MHD critical point and expand to zero pressure at infinite radial distance from the source. Because the constants of motion must be specified to extremely high accuracy, the critical solutions cannot be found analytically. Synopsis of many numerical solutions reveals a uniform parametric characterization of the asymptotic wind in terms of one combination of source parameters,Z, the mean source particle energy divided by mc2[sgrave]½, where [sgrave] is a generalization of Michel's (1969) cold relativistic wind strength parameter. Cool winds, withZ<1, behave asymptotically much as Michel's cold wind minimum torque solution;Z1 hot winds have quite different, but simply characterized, asymptotic solutions. Thus, the strength of magnetized relativistic outflows can depend critically upon the temperature of the source.