A category enriched in the two-element lattice is a preordered set

Let $\mathcal{A}$ be $\mathbf{2}$-enriched. Define a relation on its object set by $A\le B⟺\mathcal{A}(A,B)=1$. Because the unit object of the base is $1$, the identity morphism of [L2] forces $\mathcal{A}(A,A)=1$ for every $A$, so the relation is reflexive. Since composition in the base is $\wedge$, the composition morphism $\mathcal{A}(B,C)\wedge \mathcal{A}(A,B)\to \mathcal{A}(A,C)$ implies that if both hom-objects on the left are $1$, then $\mathcal{A}(A,C)=1$ on the right. So the relation is transitive. By [L1], it is a preorder.

Conversely, given a preorder $(P,\le )$, put the object set equal to $P$ and define the hom-object by $\mathcal{A}(x,y)=1$ when $x\le y$ and $\mathcal{A}(x,y)=0$ otherwise. Reflexivity gives the identity maps, and transitivity gives the composition morphism because $1\wedge 1=1$ exactly in the composable case. Thus the preorder data satisfy [L2].

The two constructions are inverse restatements of the same information, so $\mathbf{2}$-enrichment and preorder structure are equivalent.