For the disjoint union, I don't like \amalg and \coprod very much. So, I did all my best to create a \varamalg with this code:
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{mathtools,amssymb}
\newcommand{\varamalg}{\sqcup\hspace{-1.085em}\raisebox{-.195em}{\rotatebox[origin=c]{37}{$\smallsetminus$}}}
\begin{document}
$A\varamalg B=A\cup B$
\end{document}
And I get:

Well, not bad for me. I used \smallsetminus because the ends are rounded.
But now, I'm not able to create \varcoprod. Could you help me?
roundrule.styat http://tex.stackexchange.com/questions/161297/is-there-such-a-thing-as-a-mathrule-rounded-endcaps/161309#161309 – Steven B. Segletes Mar 18 '14 at 16:56\varcoprodyou're just after an operator-type\varamalg? If not, could you explain more what\varcoprodshould look like, and how you plan on using it? – Werner Mar 18 '14 at 17:14