The following example uses \uparrow
as base unit, because the arrow reaches from the bottom to the top of the bounding box without interfering side bearings.
Macro \@leftuparrow
takes three arguments with values 0
and 1
each to denote the left, northwest, and up arrow component of the total symbol.
As the examples show, a font for the up arrow should be used, where the tip is not too broad.
\documentclass{article}
\usepackage{amsmath}
%\usepackage{MnSymbol}
%\usepackage{mathabx}
\usepackage{graphicx}
\makeatletter
\newcommand*{\leftuparrow}{%
\@leftuparrow101%
}
\newcommand*{\loweredleftarrow}{\@leftuparrow100}
\newcommand*{\leftnwarrow}{\@leftuparrow110}
\newcommand*{\leftnwuparrow}{\@leftuparrow111}
\newcommand*{\nwuparrow}{\@leftuparrow011}
\newcommand*{\@leftuparrow}[3]{%
\mathrel{%
\mathpalette{\@@leftuparrow{#1}{#2}{#3}}{}%
}%
}
\newcommand*{\@@leftuparrow}[5]{%
% #1: left
% #2: nw
% #3: up
% #4: math style
% #5: unused
\sbox0{$#4\uparrow\m@th$}% base unit
%
% Correction of depth to compensate line width
\sbox6{$#4\mkern.25mu\m@th$}%
\dimen@=\dp0 %
\advance\dimen@ by -\wd6 %
\dp0=\dimen@
%
\sbox2{%
\rotatebox[origin=b]{45}{\hbox to\z@{\hss\copy0\hss}}%
}%
\sbox4{%
\rotatebox[origin=b]{90}{\copy0}%
}%
\dimen@=\z@
\dimen6=\z@
\ifnum#1#2>\z@
\kern4\wd6 % left side bearing
\fi
\ifnum#1=1 % leftarrow
\copy4 %
\dimen@=\wd4 %
\dimen6=\wd6 %
\fi
\ifnum#2=1 % nwarrow
\ifdim\dimen@>\z@
\llap{\copy2}%
\else
\copy2 %
\dimen@=\wd2 %
\dimen6=.7071\wd6 %
\fi
\fi
\ifnum#3=1 % uparrow
\ifdim\dimen@>\z@
\kern-.5\wd0 %
\fi
{\uparrow}%
\else
\kern4\wd6 % right side bearing
\fi
\kern\dimen6 %
}
\makeatother
\begin{document}
\newcommand*{\test}{%
\loweredleftarrow \leftnwarrow \leftnwuparrow
\leftuparrow \nwuparrow \uparrow
}
\setlength{\fboxsep}{0pt}%
\setlength{\fboxrule}{.1pt}%
\begin{gather*}
\fbox{$\leftarrow$}
\fbox{$\uparrow$}
\fbox{$\nwarrow$}
\\
\fbox{$\loweredleftarrow$}
\fbox{$\leftnwarrow$}
\fbox{$\leftnwuparrow$}
\fbox{$\leftuparrow$}
\fbox{$\nwuparrow$}
\\
\test\\
\scriptstyle\test\\
\scriptscriptstyle\test
\end{gather*}
\end{document}
Result for Computer Modern fonts:

Result for MnSymbol
:

Result for mathabx
:
