Cokernel

Remarks

Equivalently, $q$ is a coequalizer of $f$ and $0 \colon X \to Y$; it is the kernel of $f$ in the opposite category. A cokernel is unique up to a unique isomorphism, is written $Y \to \operatorname{coker} f$, and is an epimorphism.

Example: in the category of modules over a ring, the quotient map $q \colon Y \to Y / f(X)$ is a cokernel of $f$. If $u \circ f = 0$, then $u$ vanishes on $f(X)$ and so induces a unique $v \colon Y / f(X) \to T$ with $u = v \circ q$. For multiplication by $n \ge 1$ on the abelian group $\Z$, the cokernel is the reduction $\Z \to \Z / n\Z$.

Non-example: for multiplication by $2$ on $\Z$, the map $\Z \to 0$ satisfies clause 1 but not clause 2. The reduction $u \colon \Z \to \Z/2$ satisfies $u \circ 2 = 0$ but does not factor through the zero group, since $u \ne 0$.