A Cat-enriched category is exactly a strict 2-category with a set of objects and small hom-categories

Assume first that $\mathcal{A}$ is enriched in $\mathbf{C}\mathbf{a}\mathbf{t}$. By [L1], each hom-object $\mathcal{A}(A,B)$ is a small category, and the object set of $\mathcal{A}$ is already a set. Because the monoidal product in [L3] is the cartesian product, the enriched composition morphism is a functor $\mathcal{A}(B,C)\times \mathcal{A}(A,B)\to \mathcal{A}(A,C)$, which is exactly horizontal composition on 1-morphisms and 2-morphisms. The identity morphism $\mathbf{1}\to \mathcal{A}(A,A)$ picks out an object of the hom-category, hence an identity 1-morphism. The enriched associativity and unit diagrams are therefore exactly the strict 2-category axioms of [L2].

Conversely, let $\mathcal{K}$ be a strict 2-category with a set of objects and small hom-categories. Use the hom-categories $\mathcal{K}(A,B)$ as the hom-objects. The horizontal-composition functor of [L2] supplies $\mathcal{K}(B,C)\times \mathcal{K}(A,B)\to \mathcal{K}(A,C)$, and each identity 1-morphism gives a functor $\mathbf{1}\to \mathcal{K}(A,A)$. Since the tensor product in [L3] is cartesian product, these data satisfy the definition of [L1].

Steps 1.1 and 1.2 are inverse unpackings of the same data, so the two notions agree exactly.