Define a structure $\mathcal{A}$ as homogeneous if, for every two substructures $\mathcal{B}, \mathcal{C}$ of $\mathcal{A}$, if $f: B \to C$ is an isomorphism, then $f$ extends to an automorphism of $\mathcal{A}$. Define a structure $\mathcal{A}$ as $\lambda$-categorical if there is, up to isomorphism, exactly one model $\mathcal{A}'$ of cardinality $\lambda$ satisfying $\mathrm{Th}(\mathcal{A})$. A structure is totally categorical if it's categorical in every infinite cardinal.
Question: is there any relation between homogeneity and (total) categoricity? In particular, does homogeneity imply (total) categoricity? If not, what is an example of a homogeneous structure which is not (totally) categorical?
I've been searching a lot for the answer for these question but without much success. I'm trying to understand these two concepts, so any help here would be greatly appreciated.
Take $(\mathbb N,0,S)$. It is vacuously homogenous because it doesn't have any proper substructures.
On the other hand, it is not even $\aleph_0$-categorical -- the usual compactness argument will produce an elementarily equivalent countable model with a nonstandard element, which cannot be isomorphic to $\mathbb N$.
If vacuous homogeneity feels like cheating, consider $(\mathbb Z,S)$ instead. This is more meaningfully homogeneous, and we can still use compactness to produce an elementarily equivalent countable structure that contains two elements that are not any finite distance from each other.