What is the generalization of "left" and "right" properties?

Question

I was looking at the usual way one proves that inverses in a group are unique, namely by supposing that they aren't and $g'$ and $\hat{g}$ are both inverses of $g$. Then

$$ g'=e*g'=(\hat{g}*g)*g'=\hat{g}*(g*g')=\hat{g}*e=\hat{g} $$

This actually presupposes only that $\hat{g}$ is a left inverse and $g'$ a right one. The same thing happens with maps, where proving a map is bijective requires only that it is both surjective and injective. A similar thing is true of the way $\leq$ and $\geq$ and $\subseteq$ and $\supseteq$ work. I was wondering if there was some (category-theoretic?) generalization of the way these work. I hope the question doesn't seem too soft or vague.

Answer 1

If a morphism has a section $g_1$ and a retraction $g_2$, then it is iso and $g_1 = g_2$ is its inverse (also c.f. "split mono", "split epi").

Is that what you meant?

Answer 2

If $f:A\to B$ is some arrow in some category that has a left-inverse $h$ and a right-inverse $g$ then it can be shown that $h=g$ on the way exposed in your question:$$h=h\circ\mathsf{id}_B=h\circ(f\circ g)=(h\circ f)\circ g=\mathsf{id}_A\circ g=g$$