coquotients of cocongruences

A cocongruence (or internal equivalence corelation) on an object $X$ of a category is a parallel pair $j_1, j_2 : X \rightrightarrows C$ which is jointly epimorphic, and such that for every object $T$, the image of $({-} \circ j_1, {-} \circ j_2) : \Hom(C, T) \to \Hom(X, T)^2$ is an equivalence relation. If pushouts exist (or at least the pushout displayed below), the latter condition is equivalent to the existence of morphisms $$\begin{align*} r & : C \to X & \quad \text{(coreflexivity)} \\ s & : C \to C & \quad \text{(cosymmetry)} \\ t & : C \to C \sqcup_{j_2,X,j_1} C & \quad \text{(cotransitivity)} \end{align*}$$ satisfying the following equations: $$\begin{align*} r \circ j_1 & = \id_X & r \circ j_2 & = \id_X \\ s \circ j_1 & = j_2 & s \circ j_2 & = j_1 \\ t \circ j_1 & = \iota_1 \circ j_1 & t \circ j_2 & = \iota_2 \circ j_2 \end{align*}$$ In particular, $j_1,j_2$ is a coreflexive pair. A coquotient of $j_1,j_2$ is an equalizer of $j_1,j_2$. We say that a category has coquotients of cocongruences if every cocongruence has a coquotient. Note that in the case of a category with binary copowers, the corresponding quotient objects $X + X \twoheadrightarrow C$ are also commonly referred to as cocongruences, or as internal equivalence corelations.