Let $\kappa$ be an infinite regular ordinal. Define $\Delta_\kappa$ to be the category of ordinals strictly smaller than $\kappa$ (i.e. the ordinals which are elements of $\kappa$, in von Neumann definition), with order-preserving maps as morphisms.

Question. Suppose $f\colon \alpha \to \beta$ is a monomorphism (resp. an epimorphism) of $\Delta_\kappa$. Is it true that necessarily $\alpha \leqslant \beta$ (resp. $\alpha \geqslant \beta$)?

Remark. It's easy to prove that monomorphisms are precisely injective order-preserving maps (resp. epimorphisms are precisely surjective oder-preserving maps), so let's assume that.