Change of base extends to enriched functors and natural transformations as a 2-functor

If $T:\mathcal{A}\to \mathcal{B}$ is a $\mathcal{V}$-functor, keep the same object map and apply $F$ to each hom-object map $T_{A,B}:\mathcal{A}(A,B)\to \mathcal{B}(TA,TB)$. Because [L1] changed both source and target hom-objects by $F$, the same compatibility diagrams from [L2] commute after applying $F$, so this gives a $\mathcal{W}$-functor $F_{\ast }T:F_{\ast }\mathcal{A}\to F_{\ast }\mathcal{B}$.

If $\alpha :T\Rightarrow S$ is a $\mathcal{V}$-natural transformation, compose each component ${\mathbf{1}}_{\mathcal{V}}\to \mathcal{B}(TA,SA)$ with the lax unit map ${\mathbf{1}}_{\mathcal{W}}\to F({\mathbf{1}}_{\mathcal{V}})$ and then with $F$ of the component. The enriched naturality equation from [L3] is preserved because [L1] builds the target hom-objects and compositions by the same laxity data. So this yields a $\mathcal{W}$-natural transformation $F_{\ast }\alpha$.

Identity 1-cells, composite 1-cells, identity 2-cells, and vertical and horizontal compositions are preserved strictly because the construction is objectwise and applies the same functor $F$ to every structural morphism. Hence $F_{\ast }$ is a strict 2-functor.