How do i get the Mollweide projection of a sphere?

Question

I have defined three angular functions

F1[θ_,ϕ_] := (1/2) (1 + Cos[θ] Cos[θ]) Cos[2ϕ]
F2[θ_,ϕ_] := -Cos[θ] Sin[2ϕ]
F3[θ_,ϕ_] := F1[θ,ϕ]^2 + F2[θ,ϕ]^2

So I can "project" F3[θ, ϕ] on a sphere in the following way

SphericalPlot3D[1,{θ,0,π}, {ϕ,0,2π}, ColorFunction -> Function[{x,y,z, θ,ϕ,r}, ColorData["Rainbow"][F3[θ,ϕ]]], ColorFunctionScaling -> False, Mesh -> True, Boxed -> True,  Axes -> True, AxesLabel -> {x, y, z}, PlotPoints -> 100, PlotLegends -> BarLegend["Rainbow"]]

and use the function to properly color the sphere. Is there a way to do the same thing using a Mollweide projection instead of a sphere?