The free enriched category is left 2-adjoint to the underlying-category construction

Define ${\mathcal{L}}_{\mathcal{V}}$ to have the same objects as $\mathcal{L}$ and hom-object ${\coprod }_{\mathcal{L}(A,B)}\mathbf{1}$ from $A$ to $B$. The summand indexed by $f:A\to B$ is the enriched name of the ordinary arrow $f$, the identity map of $\mathcal{L}$ names the enriched identity, and ordinary composition in $\mathcal{L}$ induces the enriched composition morphisms by the coproduct universal property. Thus [L2] gives a $\mathcal{V}$-category.

Conversely, let $F:\mathcal{L}\to {\mathcal{B}}_0$ be an ordinary functor. On objects keep the same map. On each hom-object ${\coprod }_{\mathcal{L}(A,B)}\mathbf{1}$, define the corresponding map into $\mathcal{B}(FA,FB)$ by naming the ordinary morphism $F(f)$ on the summand indexed by $f$. The functoriality of $F$ makes these maps preserve identities and composition, so [L2] gives a $\mathcal{V}$-functor ${\mathcal{L}}_{\mathcal{V}}\to \mathcal{B}$.

Let $T:{\mathcal{L}}_{\mathcal{V}}\to \mathcal{B}$ be a $\mathcal{V}$-functor. For every ordinary arrow $f:A\to B$ in $\mathcal{L}$, the corresponding coproduct summand $\mathbf{1}\to {\mathcal{L}}_{\mathcal{V}}(A,B)$ followed by the hom-map of $T$ gives a global element $\mathbf{1}\to \mathcal{B}(TA,TB)$, hence by [L1] an ordinary morphism $TA\to TB$ in ${\mathcal{B}}_0$. Compatibility of $T$ with identities and composition makes these assignments an ordinary functor $\mathcal{L}\to {\mathcal{B}}_0$.

The same correspondence acts on $2$-cells. A $\mathcal{V}$-natural transformation $\alpha :T\Rightarrow S$ has components $\mathbf{1}\to \mathcal{B}(TA,SA)$, hence ordinary components $TA\to SA$ in ${\mathcal{B}}_0$ by [L1]. Its enriched naturality equation on the coproduct hom-object of ${\mathcal{L}}_{\mathcal{V}}$ holds exactly when the ordinary naturality square holds on every summand indexed by an arrow of $\mathcal{L}$. Thus enriched natural transformations correspond bijectively to ordinary natural transformations, compatibly with identities and composition.

Steps 1.2 and 2.1 give mutually inverse correspondences on functors, while step 2.2 gives the corresponding isomorphism on morphisms. They therefore define a natural isomorphism of hom-categories $[{\mathcal{L}}_{\mathcal{V}},\mathcal{B}{]}_0\cong [\mathcal{L},{\mathcal{B}}_0],$ which is the claimed $2$-adjunction.