It is a matter of unwinding the definitions to see that for a flat (set valued) presheaf the W-weighted colimit functor

W ⊗ _

commutes with finite limits: this is basically because the W-weighted colimit is a colimit over the opposite of the category of elements of W, and since W is flat the latter is filtered. This is in turn equivalent to the request that the Yoneda extension $Yan(W)=Lan_YW$ commutes with finite limits.

This can be generalized to the case of a sound doctrine $\mathbb D$ (see [1]) to get that the following are equivalent for a weight W:

- Yan(W) is $\mathbb D$-continuous
- Elts(W)° is $\mathbb D$-filtered
- W-weighted colimits commute with $\mathbb D$-limits

Let now $\mathbb D$ be a sound doctrine: it is natural to ask whether there are conditions analogous to $\mathbb D$-flatness that ensure that W-weighted colimits commute with U-weighted limits, i.e. that there is an isomorphism

$$W ⊗ \{U, F\} \cong \{U, W⊗F\}$$

for weights W, U and $F : I \times J \to K$ a functor.

This condition stronger than $\mathbb D$-flatness is such that $\mathbb D$-flatness correspond to weighted flatness with respect to conical weights with $\mathbb D$-domain.

Is this condition non-trivial? What is an equivalent condition for weighted $\mathbb D$-flatness?

The only reference I can find for this condition is [2], where no such criterion is stated, at least at a rapid glance.

[1]: *Adámek, Jiří; Borceux, Francis; Lack, Stephen; Rosický, Jiří*, **A classification of accessible categories**, J. Pure Appl. Algebra 175, No.1-3, 7-30 (2002). ZBL1010.18005.

[2]: *Lack, Stephen; Rosický, Jiří*, **Homotopy locally presentable enriched categories**, Theory Appl. Categ. 31, 712-754 (2016). ZBL1346.18025.