Implication details

Proof: Let $\C$ be a generalized variety and let $G$ be an essentially small collection of strongly finitely presentable objects which generates $\C$ under sifted colimits. Consider $G$ as a full subcategory of $\C$ and consider the restricted Yoneda functor $$Y : \C \to [G^{\op},\Set], \quad X \mapsto \Hom(-,X)|_{G^{\op}}.$$ It preserves all limits and, since the objects of $G$ are strongly finitely presentable, all sifted colimits. By Adámek–Rosický, Lemma 3.8, $Y$ is fully faithful.Now let $q_1, q_2 : E \rightrightarrows X$ be a congruence in $\C$. In particular, it is a reflexive pair. Since $\C$ has sifted colimits, it has reflexive coequalizers. In particular, $q_1, q_2$ have a coequalizer $h : X \to Q$. Let $T \in G$. Since $Y$ preserves this coequalizer and colimits in $[G^{\op},\Set]$ are computed pointwise, the map $h_* : \Hom(T,X) \to \Hom(T,Q)$ is a coequalizer of $$(q_1)_*, (q_2)_* : \Hom(T,E) \rightrightarrows \Hom(T,X)$$ in $\Set$. By definition of a congruence, $(q_1)_*, (q_2)_*$ are jointly injective and their image in $\Hom(T,X)^2$ is an equivalence relation, i.e. they form a congruence in $\Set$. Since $\Set$ has effective congruences, this lemma shows that $(q_1)_*, (q_2)_*$ is a kernel pair of $h_*$. Since $T \in G$ was arbitrary and limits in $[G^{\op},\Set]$ are computed pointwise, $Y(q_1), Y(q_2)$ is a kernel pair of $Y(h)$. Since $Y$ is fully faithful, it reflects limits. Hence, $q_1, q_2$ is a kernel pair of $h$.Remark: For multi-algebraic categories, this result is also proven in Diers, Thm. 4.0 (see also its English translation).