Enrichment in a preorder recovers a preorder, and enrichment in sets recovers a small ordinary category

Suppose the base $\mathcal{V}$ is thin. Then for any objects $A,B$ of $\mathcal{A}$, the hom-set of the underlying ordinary category is $\mathcal{V}(\mathbf{1},\mathcal{A}(A,B))$. Because $\mathcal{V}$ has at most one morphism between any two objects, this hom-set has at most one element. The enriched identities and composition from [L3] become ordinary identities and composition, so the underlying category is a category all of whose hom-sets are subsingletons. Defining $A\le B$ when such a morphism exists gives a reflexive and transitive relation by [L2], hence a preorder by [L1].

Now take $\mathcal{V}=\mathbf{S}\mathbf{e}\mathbf{t}$ with cartesian product and singleton unit. Then a hom-object of [L3] is literally a set of morphisms, an identity map $\{\ast \}\to \mathcal{A}(A,A)$ chooses an identity element, and the composition morphism $\mathcal{A}(B,C)\times \mathcal{A}(A,B)\to \mathcal{A}(A,C)$ is ordinary composition of elements. The associativity and unit diagrams of [L3] are exactly the axioms of [L2]. Conversely, every small ordinary category gives such data by taking its hom-sets as the enriched hom-objects; its object collection is a set as required by [L3].

Therefore thin-base enrichment recovers a preorder on the object set, and Set-enrichment recovers precisely a small ordinary category.