\varprojlím and \varinjlím

You can solve it with a simple patch, which probably should be the default also for amsmath: Since babel-spanish already solves the problem for \lim, we can just use the command instead of the explicit lim in the definition of \varlim@.

\documentclass[a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[spanish]{babel}

\usepackage{amsmath}
\usepackage{etoolbox}

\makeatletter
\patchcmd{\varlim@}{lim}{\lim}{}{}
\makeatother

\begin{document}

$\displaystyle\lim\varinjlim\varprojlim$

\end{document}

This is the output after adding \usepackage[sc]{mathpazo}:

For “colimit”:

\documentclass[a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[spanish]{babel}
\usepackage[sc]{mathpazo}

\usepackage{amsmath}
\usepackage{etoolbox}

\DeclareMathOperator*{\colim}{co{\lim}}
\makeatletter
\patchcmd{\varlim@}{lim}{\lim}{}{}
\makeatother


\begin{document}

$\lim\colim\varinjlim\varprojlim$

\end{document}