ClearAll[f1, f2, f3];
f1[[Eta]_ /; [Eta] > 0] = 1.;
f1[[Eta]_ /; [Eta] == 0] = 2.;
f2[[Eta]_ /; [Eta] > 0] = 1./[Eta];
f2[[Eta]_ /; [Eta] == 0] = 0.;
f3[[Eta]_ /; [Eta] > 0] = 4./[Eta]^2 f[[Eta]];
f3[[Eta]_ /; [Eta] == 0] = 4.;

NDSolve[{f1[[Eta]] !(
*SubscriptBox[([DifferentialD]), ([Eta], [Eta])](f[[Eta]])
) + f2[[Eta]] !(
*SubscriptBox[([DifferentialD]), ([Eta])](f[[Eta]])) +
f[[Eta]] - f3[[Eta]] - f[[Eta]]^3 == 0, f[0] == 0, f[10] == 1},
f[[Eta]], {[Eta], 0, 10}]

ode=1/η D[η D[f[η], η], η]+(1 - s^2/η^2) f[η] - f[η]^3 == 0//Expand//Collect[#, f[η]]&

Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];
Module[{minη, maxη},
  Manipulate[
    sol = Quiet@NDSolve[Join[{ode/.s->s0}, {f[1] ==f1, f'[1] ==fd1}], f, {η, 0, 10}];
    {minη, maxη} = First@InterpolatingFunctionDomain[f /. sol[[1]]];
    Plot[f[η] /. sol, {η, minη, maxη}, PlotRange->{{0, 10}, {0, 1}}]
    ,{{f1, 0.520052}, 0.52, 0.5205}, {{fd1,0.410635}, 0.409, 0.412}, {{s0,1}, 0, 3, 1}
  ]
]

ode /. f -> (h[(#^2 - 1)/(#^2 + 1)]&) /. η -> Sqrt[(1 + z)/(1 - z)] // Simplify
odez = (1 + z) # & /@ %

Manipulate[
  sol2=Quiet@NDSolve[Join[{odez/.s->1},{h[4/5]==h0,h'[4/5]==hd0}],h{z,-1,1}];
  GraphicsColumn[
    {Plot[{h[z]/.sol2},{z,-1,1},PlotRange->{0,1}],
     Plot[{h[(η^2-1)/(η^2+1)]/.sol2,f[η]/.sol},{η,0,20},PlotRange->{0,1}]}
  ]
  ,{{h0,0.917477},0.90,0.94},{{hd0,0.546211827},0.5,0.6}
]

Plot[{h[z] /. sol2, (1 + z)/2}, {z, -1, 1}, PlotRange -> {0, 1}]

(1 + z)/2 /. z -> (η^2 - 1)/(η^2 + 1) // Simplify

Plot[
  {h[(η^2 - 1)/(η^2 + 1)] /. sol2,
   η^2/(η^2 + 1)}, {η, 0, 20},
  PlotRange -> {0, 1}, AxesLabel -> {η}, AspectRatio -> 0.4
]