CatDat

Implication Details

Assumptions: core-thin

Conclusions: quotients of congruences

Reason: If $p_1, p_2 : E \rightrightarrows X$ is a congruence, the symmetry morphism $s : E \to E$ is an automorphism of $E$, hence equal to $\mathrm{id}_E$ by assumption. But then $p_1 = p_2 \circ s = p_2$, and simply $\mathrm{id}_X$ is a coequalizer.