Creating a variant of \varinjlim without redefining \varinjlim itself

Question

I would like to define a command \laxcolim which renders the same as \varinjlim does, except with two "l"'s instead of one. But I still want the usual \varinjlim command to be available to me. (Similarly, I want a command \laxlim which is like \varprojlim but with an extra "l".)

I tried modifying this solution to the following:

\documentclass[ a4paper, leqno]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amssymb, amsfonts, mathtools}
\makeatletter\def\varlim@#1#2{%
\vtop{\m@th\ialign{##\cr
\hfil$#1\operator@font llim$\hfil\cr
\noalign{\nointerlineskip\kern1.5\ex@}#2\cr
\noalign{\nointerlineskip\kern-\ex@}\cr}}%
}
\makeatother
\begin{document}
[ \varinjlim( E_{α},f_{α,β}),\quad\varprojlim( E_{α},f_{β,α}),]%
\end{document}​ 

The result looks great except that it changes the \varinjlim command itself, whereas I need both the usual \varinjlim command and my modified one to be available to me.

To address this issue, I made the following changes. I don't actually know what I'm doing, but the intention is to define a new command \varllim rather than overwriting the \varlim command, and then to copy the usual definition of \varinjlim, but with \varllim used in place of \varlim.

\documentclass[ a4paper, leqno]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amssymb, amsfonts, mathtools}
\makeatletter\def\varllim@#1#2{%
\vtop{\m@th\ialign{##\cr
\hfil$#1\operator@font llim$\hfil\cr
\noalign{\nointerlineskip\kern1.5\ex@}#2\cr
\noalign{\nointerlineskip\kern-\ex@}\cr}}%
}
\makeatother
\def\laxcolim{%
  \mathop{\mathpalette\varllim@{\rightarrowfill@\textstyle}}\nmlimits@
}
\def\laxlim{%
  \mathop{\mathpalette\varllim@{\leftarrowfill@\textstyle}}\nmlimits@
}
\begin{document}
[ \varinjlim( E_{\alpha},f_{\alpha,\beta}),\quad\varprojlim( E_{\alpha},f_{\beta,\alpha}),]%
[ \laxcolim( E_{\alpha},f_{\alpha,\beta}),\quad\laxlim( E_{\alpha},f_{\beta,\alpha}),]%
\end{document}​

I get an error saying that that \varllim is an undefined control sequence. I think I am missing something very basic!

Answer 1

Rather than reimplementing the whole thing, it's simpler to do a couple of patches:

\documentclass[ a4paper, leqno]{article}
\usepackage{amsmath}
\usepackage{xpatch}
\NewCommandCopy{\laxcolim}{\varinjlim}
\makeatletter
\NewCommandCopy{\varllim@}{\varlim@}
\xpatchcmd{\laxcolim}{\varlim@}{\varllim@}{}{}
\xpatchcmd{\varllim@}{lim}{llim}{}{}
\makeatother
\begin{document}
[
\varinjlim( E_{\alpha},f_{\alpha,\beta}),
\quad
\laxcolim( E_{\alpha},f_{\beta,\alpha}),
]
\end{document}

Another possible strategy:

\documentclass[ a4paper, leqno]{article}
\usepackage{amsmath}
\makeatletter
\newcommand{\genericinjlim}[1]{%
  \mathop{\mathpalette\varlim@{{#1}{\rightarrowfill@\textstyle}}}\nmlimits@
}
\def\varlim@#1#2{\varlim@@#1#2}
\def\varlim@@#1#2#3{%
  \vtop{\m@th\ialign{##\cr
    \hfil$#1\operator@font #2$\hfil\cr
    \noalign{\nointerlineskip\kern1.5\ex@}#3\cr
    \noalign{\nointerlineskip\kern-\ex@}\cr}}%
}
\makeatother
\renewcommand{\varinjlim}{\genericinjlim{lim}}
\newcommand{\laxcolim}{\genericinjlim{llim}}
\begin{document}
[
\varinjlim( E_{\alpha},f_{\alpha,\beta}),
\quad
\laxcolim( E_{\alpha},f_{\beta,\alpha}),
]
\end{document}

You can supplement this with

\newcommand{\genericprojlim}[1]{%
  \mathop{\mathpalette\varlim@{{#1}{\leftarrowfill@\textstyle}}}\nmlimits@
}

for projective limits.

In order to get better arrows, one can use old-arrows.

\documentclass[a4paper,leqno]{article}
\usepackage{amsmath}
\usepackage[old]{old-arrows}
\makeatletter
\newcommand{\genericinjlim}[1]{%
  \mathop{\mathpalette\varlim@{{#1}{\varrightarrowfill@\textstyle}}}\nmlimits@
}
\newcommand{\genericprojlim}[1]{%
  \mathop{\mathpalette\varlim@{{#1}{\varleftarrowfill@\textstyle}}}\nmlimits@
}
\def\varrightarrowfill@{\arrowfill@\relbar\relbar\varrightarrow}
\def\varleftarrowfill@{\arrowfill@\varleftarrow\relbar\relbar}
\def\varlim@#1#2{\varlim@@#1#2}
\def\varlim@@#1#2#3{%
  \vtop{\m@th\ialign{##\cr
    \hfil$#1\operator@font #2$\hfil\cr
    \noalign{\nointerlineskip\kern1.5\ex@}#3\cr
    \noalign{\nointerlineskip\kern-\ex@}\cr}}%
}
\makeatother
\renewcommand{\varinjlim}{\genericinjlim{lim}}
\renewcommand{\varprojlim}{\genericprojlim{lim}}
\newcommand{\laxcolim}{\genericinjlim{llim}}
\newcommand{\elts}{\genericinjlim{\mathsf{elts}}}
\begin{document}
[
\varinjlim( E_{\alpha},f_{\alpha,\beta})
\quad
\laxcolim( E_{\alpha},f_{\beta,\alpha})
\quad
\varprojlim( E_{\alpha},f_{\beta,\alpha})
\quad
\elts(F)
]
\end{document}