A special kind of functor pair

Question

Assume $C, D$ be two category, $F : C \to D$, $G : D \to C$ be two functors satisfy $F \circ G \circ F = F$ and $G \circ F \circ G = G$. This kind of pair of functors has many applications. So, does this property has its own exclusive name?

Answer 1

I couldn't find an established name for this, but you can probably call $G$ a "quasi-inverse" of $F$ (and $F$ is a quasi-inverse of $G$). Though it's probably more interesting in category to have natural isomorphisms $FGF \simeq F$, $GFG \simeq G$.

Answer 2

If you loosen you equalities to isomorphisms and let $F,G$ be adjoints, then this is asking simply that both $F$ and $G$ are fully faithful.