No, it's not. 0.9 C is from the Stefan-Boltzmann relationship; that's how I found it.
Ahh, the top of the troposphere effective value that some folks use, that does not take surface albedo (i.e. surface reflection) nor does it take IR re-radiation upward into account.
In fact that flux measure is an effective value for computational purposes only (an equivalency with solar radiation intercepted by the earth.) Since CO2 is mixed throughout the atmosphere acting as a sink for IR re-radiating both upward and downward toward the suface from the atmosphere after being generated at the surface converted from visible solar radiation, and not an external source of IR in space radiating downward from the top of the atmosphere.
The IR value at the surface of 0.2oC for each doubling of CO2 concentration in the atmosphere is for the hypothesized 1.5 watts increase in surface IR flux hypothesized for CO2. Not the approximately theoretical 4 watts generally calculated as an value applied the top of the troposphere as an effective flux externally applied downward for CO2.
I suggest if you are going to look at the effect on surface tempertures, which supposedly is the value of concern in global warming effects on the Earth's surface, you should use the surface flux measure, not the top of atmosphere measure that does not implement albedo nor the re-radiation of thermal radiation upward.
Refer to #12 for the at the surface measure of 0.2oC direct radiative effects on Earth's surface temperature which is what the concern is really about as regards the hypothetical "Global Warming" effects like melting icecaps, warming oceans, extermination of species, coastal flooding wiping out population centers, destruction of mankind with massive hurricanes and violent weather from surface heating, etc. ...
4: Rethinking the Global Radiative Forcing Concept (Chapter 4 from the online National Academies Press book Radiative Forcing of Climate Change: Expanding the Concept and Addressing Uncertainties (2005)
The chart on page 88 is impressive. And it shows for CO2 that if the tropopause forcing is 1 W m-2, that the surface forcing for CO2 can range from 0.38 to 1.12 W m-2. So if both factors are multiplied by 4 for doubled CO2 (giving 4 W m-2 at the tropopause), then the surface forcing can range from ~1.6 (your preferred value) to 4.48 W m-2. Correct? This shows the wide range of uncertainties.
You might comment on the section beginning on page 89, "Global mean radiative forcing at the surface".