Embeddings imply isomorphism

Question

Suppose you have two structures $X,Y$ over the same signature $\sigma$. If you have embeddings $f:X\to Y$ and $g:Y\to X$, does this imply the existence of an isomorphism between $X$ and $Y$?

Answer 1

A simple counterexample from $L=\{<\}$: Let $X=(0,1)$ and $Y=(0,1)\cup(1,2).$ $X$ embeds in $Y$ by inclusion, and $Y$ embeds in $X$ by just shrinking $Y$ down and putting it inside $X.$ Yet these are not isomorphic (for instance, $Y$ does not have the least upper bound property).