Definition:Injective Object

Definition
Let $\mathbf A$ be an abelian category.

An object $I$ of $\mathbf A$ is an injective object, if for every monomorphism $i : X \to Y$ in $\mathbf A$ and every morphism $f : X \to I$, there exists a morphism $g : Y \to I$, such that $g \circ i = f$.