Implication details

Proof: Let $\C$ be a multi-algebraic category. We may assume $\C = \Mod(\A)$, where $\A$ is a small category equipped with a collection of finite discrete cones and discrete cocones, and $\Mod(\A) \subseteq \Hom(\A,\Set)$ consists of those functors that map the finite discrete cones in $\A$ to finite product cones in $\Set$ and the discrete cocones in $\A$ to coproduct cocones in $\Set$. Since finite products and coproducts commute with pullbacks in $\Set$, it is easy to check that $\Mod(\A) \subseteq \Hom(\A,\Set)$ is closed under pullbacks. Thus, pullbacks are constructed pointwise. Since surjective maps are stable under pullback in $\Set$, it remains to show, as in the case of finitary algebraic categories, that regular epimorphisms in $\C$ are precisely the pointwise surjective morphisms of models.

First, let $\alpha : F \to G$ be any morphism of models. For $A \in \A$, consider the image $I(A) = \im(\alpha(A)) \subseteq G(A)$. This defines a subfunctor $I \subseteq G$. It is a model because $F$ and $G$ are models and, for any family of maps of sets $(f_i : X_i \to Y_i)_{i \in I}$, we have $\im(\prod_{i \in I} f_i) = \prod_{i \in I} \im(f_i)$ and $\im(\coprod_{i \in I} f_i) = \coprod_{i \in I} \im(f_i)$. Therefore, $\alpha$ factors in $\C$ as $$\alpha = \iota \circ \pi,$$ where $\pi : F \to I$ is pointwise surjective and $\iota : I \to G$ is pointwise injective, hence a monomorphism. If $\alpha$ is a regular epimorphism, then it is in particular an extremal epimorphism, so $\iota$ is an isomorphism. Thus, $I = G$, and $\alpha$ is pointwise surjective.

Conversely, assume that $\alpha : F \to G$ is pointwise surjective. Then it is the coequalizer of its kernel pair $F \times_G F \rightrightarrows F$ in the category $\Hom(\A,\Set)$ of all functors. Since the pullback $F \times_G F$ is a model, $\alpha$ is also the coequalizer in the full subcategory of models.