conditional - Can I add different texts to these two pictures using the `If` function?

I use the code from an answer to get two pictures:

F[z_] := z^2;
t1 = 0; t2 = Pi/3; dt = (t2 - t1)/10;
r1 = 1; r2 = 3; dr = (r2 - r1)/10;

GraphicsRow[
 With[{z = r Exp[I t], col = Black},
    Show[
     ParametricPlot[
      ReIm@#[z], {r, r1, r2}, {t, t1, t2},

      Mesh -> 9,
      MeshShading -> ArrayPad[
        {{None, col}, {col, col}, {None, col}}, {{5, 2}, {4, 4}}, None
        ],
      Frame -> False,
      AxesOrigin -> {0, 0},
      PlotRange -> {{-6, 10}, {-1.1, 10}},
      ImageSize -> Medium,
      AxesStyle -> Opacity[0.1]],
     Graphics[{

       {Blue, Circle[], Text[0, {-0.2, -0.3}]},
       {Blue, Arrow[{{0, 0}, {10, 0}}]},
       {Blue, Arrow[{{0, 0}, 10 ReIm@#[ Exp[I Pi/3]]}]}
       }]
     ]
    ] & /@ {Identity, F}]

Can I add different texts to these two pictures using the If function?

[Added: To clarify my question, I'd like to add the angle symbol \[Theta] to the first figure and 2\[Theta] to the second one. Once I can do this, I'm expecting that I will be able to add other different stuffs to these two figures.]

---

I'm expecting something like If[#==Identity, Text[\[Theta], {-0.2, -0.3},[]], but I don't know the correct syntax: #==Indentity does not work.

Answer

Thanks to LouisB's comment, I can do it with replacing the Graphics[] in the code as

Graphics[{
  {Blue, Circle[], {FontSize -> 12, 
    Text[If[TrueQ[# == Identity], "\[Theta]", 
      "2\[Theta]"], {.44, .25}]}},
  {Blue, {FontSize -> 16, 
    Text[If[TrueQ[# == Identity], "\[Infinity]", ""], {10.5, 0}]}},

  {Blue, {FontSize -> 16, 
    Text[If[TrueQ[# == Identity], "\[Infinity]", ""], 
     10.5 ReIm@#[ Exp[I Pi/3]]]}},
  {Blue, Arrow[{{0, 0}, {10, 0}}]},
  {Blue, Arrow[{{0, 0}, 10 ReIm@#[ Exp[I Pi/3]]}]}
  }]