Let $\mathcal{C}$ be a category enriched in $\mathcal{V}$, and $\mathcal{D}$ be a category enriched in $\mathcal{W}$. It is well known that $\mathcal{V}Cat$, the category of $\mathcal{V}$-enriched categories, forms a $2$-category.
If we have some $2$-functor, then, say $F : \mathcal{V}Cat \to \mathcal{W}Cat$, that is (in the 2-categorical sense) left-adjoint to another $2$-functor $G : \mathcal{W}Cat \to \mathcal{V}Cat$, then does $F$ necessarily preserve colimits in the sense that the object map of $F$ takes diagrams of $\mathcal{V}$-colimits in $\mathcal{C}$ to diagrams of $\mathcal{W}$-colimits in $\mathcal{D}$?