Order in writing composed morphisms

Yes, the usual order of composition is awkward. Unfortunately, it is hard to imagine how to change this mistake. Here are some links which are related to your question:

• Category theory text that defines composition backwards?
• http://ncatlab.org/nlab/show/composition#Notation
• https://en.wikipedia.org/wiki/Reverse_Polish_notation

For example, you will find that "reverse Polish has been found to lead to faster calculations", and that "Many people who agree that diagrammatic order is 'better' on its own merits nevertheless believe that trying to change the established 'classical' order of composition creates more confusion than it removes".

By the way, I have recently written an introduction to (weak) $2$-categories with the usual composition notation, and it was really just a big mess. For instance, try to write down the interchange law without confusing yourself or the reader! I was really wondering why in their texts almost all category theorists just cope with this mess instead of changing it. Then I chose the diagrammatic order of composition and have written $f * g$ (not $f;g$ which is kind of ugly) for the composition (first $f$, then $g$). I also wrote $(x)f$ instead of $f(x)$, because then we have $(x)(f * g)=((x)f)g$. The result was unusual, but also enlightening, because now every formula and every part of a formula could be read and simultaneously evaluated from left to right, which is what we do anyway in most countries around the world.

Notice how natural $(x)f$ really is: First, I have my $x$, and then I plug this $x$ into $f$. This is how calculations work. The notation $f(x)$ is not so good because we have to put $f$ on a stack while evaluating this expression (either by our mind, or by a parser).

Actually RPN (Reverse Polish Notation) is completely free of brackets, but I would suggest to add brackets for readability and also still use infix notation for binary opaerations. I would prefer $2+3$ to $2 \,3 + $.