The definition of cokernels says that a cokernel $f : Y \rightarrow X$ is a pair $(C; c)$ of an object $C$(a cokernel
object) and a morphism $c : X \rightarrow C $(a cokernel morphism) such that $c$
(1) composes with $f$ to the zero morphism : $c \circ f = 0_{YC}$ and
(2) is a universal such morphism in the sense that any other such morphism $d: X \rightarrow D$ factors through c
as $d = g \circ c$ for some unique morphism $g : C \rightarrow D$.
I understand that C has to be the coker $f =\frac{x}{im(f)}$ but I am not sure what map $c$ should be and also, how should I define the map $g$?
Best Answer
If $C$ is given by the quotient $\def\im{\operatorname{im}}X/\im f$, then the only natural map $c:X\to C$ is the projection $X\twoheadrightarrow X/\im f$. Certainly every $x\in\im f$ will vanish under this map.
Moreover, if $d:X\to D$ vanishes on $\im f$, then the first isomorphism theorem (or what is apparently called the "fundamental theorem on homomorphisms" but in any case) this factors uniquely as $X\twoheadrightarrow X/\im f\to D$.