Ok, so how do you remember which are the epimorphisms, which are the monomorphisms, and which way around the funny arrows get used?
Since the textbooks don’t seem eager to offer helpful mnemonics, I offer a forgetful world the following.
It’s the LM/PR rule. L-for-left goes with M-for-mono, and P-almost-for-epi goes almost next to R-for-right. OK?
But what does that mean? Simple. A mono is of course a left-cancellable morphism, and you signal one using an arrow with an extra decoration (a tail) on the left. Dually, an epi is a right-cancellable morphism, and you signal one of those using an arrow with an extra decoration (another head) on the right.
Easy, huh? Well, it works for me — and these days, I’m grateful for all the props I can get … [As always, click on the image to get a full sized version.]