Is there a name for this structure?

[Michael Ackerman]

Joe English wrote:

> Suppose you have two morphisms f : A -> B and g : B -> A
> such that neither (f . g) nor (g . f) is the identity,
> but satisfying (f . g . f) = f.   Is there a conventional name
> for this?  Alternately, same question, but f and g are functors
> and A and B categories.
> 
> In some cases (g . f . g) is also equal to g; is there a name
> for this as well?

I believe there isn't really a standard name for this, as evidenced by
the following.
In Mac Lane's "Categories for the Working Mathematician", p 21 of 1st or
2nd edn, in
an exercise he defines "an arrow f:a ->b in a category C is _regular_
when there exists an arrow g: b -> a such that f g f = f". But this
usage is highly non-standard; in standard usage there are regular
epimorphisms (and regular categories defined in terms of them) but
they're rather more involved.

I think I've seen it said that f is a quasi-inverse of g (or is it the
other way round?), but I can't find a reference.

-- Michael Ackerman