# Do 2-categorical left adjoints preserve $\mathcal{V}$-colimits?

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 $\mathcal{D}$?

• What do you mean by "takes diagrams of $\cal V$-colimits to diagrams of $\cal W$-colimits"? – Fosco Loregian Sep 10 '17 at 15:48
• @FoscoLoregian To be honest, I'm not quite sure. I assumed there would be an a theory analogous to the usual theory of diagram categories in standard category theory in enriched category theory, but perhaps this is not the case when one considers graded limits? In any case, my question I think could be reasonably interpreted (and generalized) as "Do 2-functors with a left adjoint (in the 2-categorical sense) preserve (weighted) 2-colimits?" I'd be interested to know the result for both strict and weak notions of 2-category (and 2-functor). – Nathan BeDell Sep 27 '17 at 22:40