Quotient of an abelian category

In mathematics, the quotient of an abelian category A by a Serre subcategory B is the category whose objects are those of A and whose morphisms from X to Y are given by the direct limit $\varinjlim \mathrm {Hom} _{A}(X',Y/Y')$ over subobjects $X'\subseteq X$ and $Y'\subseteq Y$ such that $X/X',Y'\in B$. The quotient A/B will then be an Abelian category, and there is a canonical functor $Q\colon A\to A/B$ sending an object X to itself and a morphism $f\colon X\to Y$ to the corresponding element of the direct limit with X'=X and Y'=0. This Abelian quotient satisfies the universal property that if C is any other Abelian category, and $F\colon A\to C$ is an exact functor such that F(b) is a zero object of C for each $b\in B$, then there is a unique exact functor ${\overline {F}}\colon A/B\to C$ such that $F={\overline {F}}\circ Q$.