Subquotient of an object in a category I would like to know the definition of "subquotient" in category theory.
Let $C$ and $D$ be categories and let $F: C\to D$ be a functor. Then I want to know what it means by "an object $Y$ in $D$ is a subquotient of $F(X)$ for an object $X$ in $C$".
I checked the Wikipedia but it did not help me.
 A: A subobject of $A$ is a (n isomorphism class of) monomorphism(s) $f:B\to A$. A quotient is a (n isomorphism class of) epimorphism(s) $g:B\to C$. So a subquotient is a quotient of a subobject of $A$, often equivalently, a subobject of a quotient.
Regarding all the parentheses: monomorphisms $f:B\to A,f':B'\to A$ represent the same subobject if there's an isomorphism $g:B\to B'$ with $f'\circ g=f$, and dually for quotients. This is really critical, for instance, for subobjects to correspond well with subsets in the category of sets, but also could slightly obscure the basic idea.
Regarding the "often," given a quotient $g:B\to C$ of a subobject $f:B\to A$ of $A$, if there are pushouts then we get an epimorphism $g':A\to D$ by pushing out $f$ along $g$, and if $f':B\to D$ is a monomorphism then we've represented $C$ as a subobject of a quotient of $A$. In many important categories, $f'$ is guaranteed to be a monomorphism, but this is not automatic.
I don't know that there's an accepted notion of subquotient when the two possible definitions don't coincide.
