Problem with plot of two y axes

Question

How to move this Y2 axis more to the right so it look better? Also how can I add lable Y1, Y2 like this?

plot1 = Plot[{ConditionalExpression[2 - x, x <= 1], 
    ConditionalExpression[x, x >= 1]}, {x, 0, 5}, 
   Frame -> {True, True, True, False}, 
   FrameStyle -> {Directive[Black, Thickness[0.003]], 
     Directive[Red, Thickness[0.003]], Automatic, Automatic}, 
   PlotStyle -> Directive[Red, Thickness[0.01]], FrameLabel -> x];
plot2 = Plot[{x/(x + 1), 1 - x}, {x, 0, 5}, 
   PlotRange -> {All, {0, 1}}, Frame -> {False, False, False, True}, 
   FrameTicks -> {{None, All}, {None, None}}, 
   FrameStyle -> {Automatic, Automatic, Automatic, 
     Directive[Blue, Thickness[0.003]]}, 
   PlotStyle -> Directive[Blue, Thickness[0.01]]];
Overlay[{plot1, plot2}]

Using the method @mrtydn mentioned I got this.
How can I change the tick number and axis label to red and blue to match the axes's colors?
How to remove the white space between the gray background and the y axis here?

    tickX = ({#1, 
      MaTeX[#1, "DisplayStyle" -> False, Magnification -> 2]} &) /@ 
   Range[0, 5, 1];
tickY1 = ({#1, 
      MaTeX[#1, "DisplayStyle" -> False, Magnification -> 2]} &) /@ 
   Range[0, 5, 1];
tickY2 = ({#1, 
      MaTeX[#1, "DisplayStyle" -> False, Magnification -> 2]} &) /@ 
   Range[0, 1, 0.2];
background1 = 
  ListLinePlot[{{0, 1}, {1, 1}, {5, 5}}, Filling -> Top, 
    FillingStyle -> Opacity[0.75, LightGray]] /. _Line -> Sequence[];
plot1 = Show[background1, 
   Plot[{ConditionalExpression[2 - x, x <= 1], 
     ConditionalExpression[x, x >= 1]}, {x, 0, 5}
    , PlotStyle -> Directive[Red, Thickness[0.01]]],
   GridLines -> Automatic,
   Frame -> {True, True, True, False}, 
   FrameStyle -> {Directive[Black, Thickness[0.005]], 
     Directive[Red, Thickness[0.005]], Automatic, Automatic}, 
   FrameLabel -> {MaTeX["\\mathrm{X}", Magnification -> 2], 
     MaTeX["\\mathrm{Y1}", Magnification -> 2], "", 
     MaTeX["\\mathrm{Y2}", Magnification -> 2]}, 
   FrameTicks -> {{tickY1, tickY2}, {tickX, None}}
   ];
plot2 = Plot[{x/(x + 1), 1 - x}, {x, 0, 5}, 
   PlotRange -> {All, {0, 1}}, Frame -> {True, True, True, False}, 
   FrameStyle -> {Directive[Black, Thickness[0.005]], 
     Directive[Blue, Thickness[0.005]], Automatic, Automatic}, 
   PlotStyle -> Directive[Blue, Thickness[0.01]], 
   FrameLabel -> {MaTeX["\\mathrm{X}", Magnification -> 2], 
     MaTeX["\\mathrm{Y2}", Magnification -> 2], "", 
     MaTeX["\\mathrm{Y2}", Magnification -> 2]}, 
   FrameTicks -> {{tickY2, ""}, {tickX, None}}];
ResourceFunction["CombinePlots"][plot1, plot2, "AxesSides" -> "TwoY"]