quotients of congruences

A congruence (or internal equivalence relation) on an object $X$ of a category is a parallel pair $q_1, q_2 : R \rightrightarrows X$ which is jointly monomorphic, and such that for every object $T$, the image of $(q_1 \circ {-}, q_2 \circ {-}) : \Hom(T, R) \to \Hom(T, X)^2$ is an equivalence relation. If pullbacks exist (or at least the pullback displayed below), the latter condition is equivalent to the existence of morphisms $$\begin{align*} r & : X \to R & \quad \text{(reflexivity)} \\ s & : R \to R & \quad \text{(symmetry)} \\ t & : R \times_{q_2,X,q_1} R \to R & \quad \text{(transitivity)} \end{align*}$$ satisfying the following equations: $$\begin{align*} q_1 \circ r & = \id_X & q_2 \circ r & = \id_X \\ q_1 \circ s & = q_2 & q_2 \circ s & = q_1 \\ q_1 \circ t & = q_1 \circ p_1 & q_2 \circ t & = q_2 \circ p_2 \end{align*}$$ In particular, $q_1,q_2$ is a reflexive pair. A quotient of $q_1,q_2$ is a coequalizer of $q_1,q_2$. We say that a category has quotients of congruences if every congruence has a quotient. Note that in the case of a category with binary powers, the corresponding subobjects $R \hookrightarrow X \times X$ are also commonly referred to as congruences, or as internal equivalence relations.