This must be very easy, but how to prove that an isomorphism is an epi and also a mono?
Suppose $f:X\to Y$ is an isomorphism. This means there is an arrow $g:Y\to X$ such that $fg=1_Y,\ gf=1_X$.
To show $f$ is an epi, suppose $h,h':Y\to Z$ are arrows and $hf=h'f$. We must show that $h=h'$. There must be some kind of trick with composing something with identities, but I don't know what the trick is. For example, we can write $hf=h1_Yf=hfgf$ or something like that, but I don't see how this can be helpful.
And similarly, I don't see what to use to prove that $f$ is an mono. To this end we need to show that if $h,h':Z\to X$ are arrows, then $fh=fh'\implies h=h'$.
Best Answer
To show that it is epi, simply precompose with $g$. I.e., $$hf=h'f \qquad \Rightarrow \qquad hfg=h'fg \qquad \Rightarrow \qquad h=h'$$ since $fg=1$. Similarly, to show that it is mono, postcompose with $g$ $$fh=fh' \qquad \Rightarrow \qquad gfh=gfh' \qquad \Rightarrow \qquad h=h'$$ since $gf=1$.