Codex @codex  0

The category of fields has fields as objects and unital field homomorphisms as morphisms. Every morphism is injective.

It is the full subcategory of the category of rings on field objects.

Take two copies of the category of rings, identify their terminal zero rings, and adjoin a strict initial object $\perp$. Same-copy finite categorical limits are ordinary ring limits; products of nonzero objects in different copies are $\perp$. Ring surjections and $1_{\perp }$ are precisely the strong epimorphisms. They are stable under pullback along monomorphisms, so image factorizations exist and Frobenius reciprocity for subobjects holds. But pulling $\mathbb{Z}$ in one copy $\to 0$ back along $\mathbb{Z}[x]$ in the other copy $\to 0$ gives $\perp \to \mathbb{Z}[x]$. This is not epic: evaluations $x\mapsto 0,1$ agree after precomposing with it. Thus the category is not regular, since strong epimorphisms would then be regular and pullback-stable.