Let $\mathcal{C}$ be a bicartesian category. It is well-known that the coproduct $\amalg :\mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ and the product $\times :\mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ are left- and right-adjoint, respectively, to the diagonal functor $\Delta_{\mathcal{C}} : \mathcal{C} \rightarrow \mathcal{C} \times \mathcal{C}$. Thus, coporoducts preserve epimorphisms and products preserve monomorphisms.
My question is the following. Is there a "large class" of categories in which coporoducts preserve monomorphisms and products preserve epimorphisms (in which the opposite statement holds)? Here by "large" I really mean that the underlying condition is not too restrictive.
Let us say that a bicartesian category $\mathcal{C}$ is split balanced if all monomorphism and epimorphisms in $\mathcal{C}$ are split. It is clear that every split balanced category has such a property.
Another such class of categories is given by bicartesian categories in which the coproduct $\amalg :\mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ and the product $\times :\mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ are surjective fully faithful functors.
However, both such conditions are too restrictive.
Best Answer
A regular category has products and pullbacks, every morphism factoring as an extremal epimorphism followed by a monomorphism, and extermal epimorphisms being stable under pullback. In particular, every epimorphism factors as an extremal (and also regular and also pullback-stable) epimorphism followed by a monomorphism that is an epimorphism.
A category is balanced if the only morphisms that are both monomorphism and a epimorphisms are isomorphisms. Thus in a balanced regular category all epimorphisms are pullback stable and in particular stable under products.
Since abelian categories are regular and balanced and self-dual, their monomorphisms are stable under coproducts, and their epimorphisms are stable under products.
An adhesive category has pushouts along monomorphisms that are themselves pullbacks and stable under pullbacks. In such a category monomorphisms are in fact regular and stable under pushouts. In particular, adhesive categories are balanced, i.e. a monomorphism that is an epimorphism is an isomorphism.
Thus, adhesive regular categories with initial object have monomorphisms stable under coproducts and epimorphisms stable under products.
Toposes are the original example of adhesive regular categories, e.g. the category of Sets. However, the category of pointed sets is also regular and adhesive with an initial object, but is not a topos.