Franklin's Notes

Linear order with successor

In a linear order $A$, we say that $x'$ is the successor of $x$ if $x<x'$ and there does not exist $y\in A$ such that $x<y<x'$. We say that $A$ has successors if every element of $A$ has a successor, and similarly we say that an order type has successors if every order of that type has successors. For instance, the order types $\omega$ and $\zeta$ have successors, and every well-ordering has successors.

We may define a subcategory $\mathsf{SLinOrd}\subset\mathsf{LinOrd}$ consisting of the linear orders that have successors, and the morphisms $f:A\to B$ preserving successors, i.e. such that $f(\mathrm{S}x)=\mathrm{S}f(x)$. Is this a reflective subcategory ?