Bimorphism which is neither injective nor surjective

Question

In Ring the inclusion $f:\mathbb Z \to \mathbb Q$ is a not surjective but injective bimorphism.

In Div the quotient map $g:\mathbb Q \to \mathbb Q / \mathbb Z$ is a not injective but surjective bimorphism.

So the question arises: Is there an algebraic category with a bimorphism which is neither injective nor surjective?

Answer 1

Can't you just form the product category of the two examples you already gave?

I suggest this as a question, because I don't really know for certain that a pair of bimorphisms from the two categories is a bimorphism in the product category, but I suspect it is true. I also don't know if such a product is considered an "algebraic category."