Looking for a disjoint symbol

Is this what you want?

documentclass{article}
usepackage{amsmath,pict2e}

makeatletter
newcommand{disjoint}{}% for safety
DeclareRobustCommand{disjoint}{%
  mathrel{%
    mathpalettedisjoint@relax
  }%
}

newcommand{disjoint@}[2]{%
  begingroup
  setlength{unitlength}{disjoint@height{#1}}%
  begin{picture}(1.4,2)
  roundcap
  linethickness{disjoint@thickness{#1}}
  put(0.7,0.5){circle{1}}
  Line(0.7,1)(0.7,2)
  end{picture}%
  endgroup
}
newcommand{disjoint@height}[1]{%
  0.8fontdimen5
  ifx#1displaystyletextfontelse
  ifx#1textstyletextfontelse
  ifx#1scriptstylescriptfontelse
  scriptscriptfontfififi 3
}
newcommand{disjoint@thickness}[1]{%
  fontdimen8
  ifx#1displaystyletextfontelse
  ifx#1textstyletextfontelse
  ifx#1scriptstylescriptfontelse
  scriptscriptfontfififi 3
}
makeatother

begin{document}

fboxsep=0ptfbox{$disjoint$}

$Adisjoint B$ 
$scriptstyle Adisjoint B$
$scriptscriptstyle Adisjoint B$

end{document}

There's a VERTICAL LINE WITH CIRCLE BELOW (U+2AF0) Unicode character and you can access to it via XeLaTeX or LuaLaTeX with a proper font:

documentclass{standalone}
usepackage{unicode-math}
setmathfont{XITS Math}
begin{document}
$⫰$ {Large $⫰$} {Huge $⫰$}
end{document}

As @Noone pointed out, there is a similar unicode character (U+2AF0). As for most unicode characters, it is included in the STIX font. It can be accessed without loading the whole font as in the following example.

documentclass{article}
DeclareFontEncoding{LS1}{}{}
DeclareFontSubstitution{LS1}{stix}{m}{n}
DeclareSymbolFont{symbols4}{LS1}{stixbb}{m}{it}
DeclareMathSymbol{midcir}{mathrel}{symbols4}{"BE}
begin{document}
( midcir )
end{document}

Using the multimap symbol...rotated using also scalerel package.

documentclass[12pt]{article}
usepackage{amsmath,amssymb}
usepackage{graphicx}
newcommand{circmap}{raisebox{.7depth}{$rotmap$}}
newcommand{rotmap}{rotatebox[origin=c]{-90}{$multimap$}}
usepackage{scalerel}
usepackage{parskip}

begin{document}
Euler's formula is remarkable: $e^{ipi} + 1 = 0$.

But if I have two sets with $A mathbin{scaleobj{.7}{circmap}} B$ we can write $A_{scalebox{.5}{$scriptscriptstyle circmap$}}^{scalebox{.5}{$scriptscriptstyle circmap$}}$

end{document}

Or another alternative can be to use fontawesome5 with the command faNeuter.

documentclass[12pt]{article}
usepackage{amsmath,amssymb}
usepackage{fontawesome5}
usepackage{graphicx}
newcommand{circmap}{raisebox{.6depth}{$rotmap$}}
newcommand{rotmap}{rotatebox[origin=c]{180}{faNeuter}}
usepackage{scalerel}

begin{document}
Le be $circmap$ a map: we define
[circmap colon A to B] 
if 
[A_{scalebox{.5}{$scriptscriptstyle circmap$}}B]

end{document}