In a category $C$ with finite colimits, a quotient object coclassifier is an object $Q$ with an epimorphism$\epsilon:Q \to \mathbb{0}$ into the initial object$\mathbb{0}$, such that for every epimorphism$f:X \to U$ there is a unique morphism $\digamma_U:Q \to X$ such that there is a pushoutdiagram of the form

$\array{
Q
&\overset{\epsilon}{\longrightarrow}&
\mathbb{0}
\\
\big\downarrow {}^{\mathrlap{\digamma_U}}
&&
\big\downarrow {}^{\mathrlap{\exists !}}
\\
X &\underset{f}{\longrightarrow}& U
}
\,.$