Coequalizer

Remarks

A coequalizer is the colimit of the diagram formed by $f$ and $g$, and it is an equalizer in the opposite category. It is an epimorphism, by the dual of the argument for equalizers.

Example: in the category of sets, let $\sim$ be the smallest equivalence relation on $Y$ with $f(x) \sim g(x)$ for all $x \in X$, let $Q = Y/{\sim}$ and let $q$ be the quotient map. Clause 1 holds by construction. If $k \circ f = k \circ g$, the relation "$k(y) = k(y')$" is an equivalence relation containing all pairs $(f(x), g(x))$, so it contains $\sim$, and $u([y]) = k(y)$ is well defined; it is unique because $q$ is surjective. In the category of abelian groups the coequalizer of $f$ and $g$ is $Y / \im(f - g)$ with the quotient map.

Non-example: for $X = \{*\}$, $Y = \{0, 1, 2\}$, $f(*) = 0$ and $g(*) = 1$, the constant map $q \colon Y \to \{*\}$ satisfies clause 1 but not clause 2. The map $k \colon Y \to \{0, 2\}$ with $k(0) = k(1) = 0$ and $k(2) = 2$ has $k \circ f = k \circ g$ but does not factor through a one-point set.