Implication details

Proof: If $q_1, q_2 : R \rightrightarrows X$ is a congruence, the symmetry morphism $s : R \to R$ is an automorphism of $R$, hence equal to $\id_R$ by assumption. But then $q_1 = q_2 \circ s = q_2$, and simply $\id_X$ is a coequalizer. Also, for the reflexivity morphism $r : X \to R$, we have $q_1 \circ r = \id$. For the reverse composition, $q_1 \circ r \circ q_1 = q_1 \circ \id$ and $q_2 \circ r \circ q_1 = q_2 \circ \id$, so since $q_1, q_2$ are jointly monomorphic, we get $r \circ q_1 = \id$. Therefore, $q_1 = q_2$ is an isomorphism, so $R$ is the kernel pair of $\id_X$.