I have three questions about when you can show there is an isometry between metric spaces.
(1) If there is an injective non-expanding map from $X$ to $Y$ and an injective non-expanding map from $Y$ to $X$, are $X$ and $Y$ isometric?
I think the answer must be no, just let $X=[0,1]$ and $Y=[0,1/2]$ with the Euclidean metric on each and let the morphisms just shrink each of the intervals by a 1/2. But $X$ and $Y$ are not isometric as metric spaces. The only reason I ask is that this question seems to imply that this is true for compact metric spaces. So maybe I am just missing something.
(2) If there is an isometric embedding from $X$ to $Y$ and an isometric embedding from $Y$ to $X$ is it true that $X$ and $Y$ are isometric?
Here by an isometric embedding I mean a map that preserves the metric.
(3) If the answer to (2) is yes, is there something to be said about which concrete categories this result holds for, with respect to embeddings?
Here I am taking the definition of concrete categories and embeddings from Adámek, Herrlich, Strecker.
I know this question sounds a lot like this question, but unless I am confused, they are talking about injective maps (monomorphisms) which make sense in any category, whereas I am talking about embeddings which are only defined for concrete categories.
EDIT: Edited to remove jargon and make clearer.
Thanks very much for any information.