TITLE: $fgf = f$, $gfg = g$, $fg$ not necessarily identity, what is this called?
QUESTION [13 upvotes]: A very simple question, I just totally forgot how it was called, and Google is not helping.
There's a pair of functions $f:X\to Y$, $g:Y\to X$.
$fgf = f$, $gfg = g$, but $fg$ and $gf$ don't need to be identities (and usually are not in interesting cases).
A simple example would be $f(a,b,c)=(a,b)$, $g(a,b)=(a,b,0)$
What were $f$ and $g$ called?

REPLY [4 votes]: This is also naturally related to adjointness: 
if you consider both $X$ and $Y$ to be poset categories, and add the conditions 

both $f$ and $g$ order-preserving 
$x \leq_X g(f(x))$ and $f(g(y)\leq_Y y$ for all $(x,y)\in X\times Y$

to your conditions (this is admittedly a little less general, 
but satisfied in many natural situations), then: 
the conditions are met 
$\Longleftrightarrow$
$f:X\leftrightarrow Y:g$ 
is an adjunction.