In the setting of $1$-categories, there are two (unweighted) variant notions of limits, namely limits and colimits. For bicategories, there are fourteen of them:

Here $\mathsf{LaxCones}(\Delta_{X},D)\overset{\mathrm{def}}{=}\mathsf{LaxNat}(\Delta_{X},D)$, and similarly for the other entries in the last column. (Incidentally, these notions are also interrelated: for instance, passing to the weighted case, weighted lax co/limits can be expressed as weighted $2$-co/limits.)

Each of these notions has an associated "strength", given by whether we require an equivalence of categories (e.g.) $$ \mathsf{Hom}_{\mathcal{D}}(X,\mathsf{bilim}^{\mathsf{lax}}(D)) \overset{\mathrm{eq}}{\cong} \mathsf{LaxCones}(\Delta_{X},D) $$ or an isomorphism of categories (e.g.)

$$ \mathsf{Hom}_{\mathcal{D}}(X,\mathsf{lim}^{\mathsf{lax}}(D)) \cong \mathsf{LaxCones}(\Delta_{X},D). $$

Bicategorical adjunctions, on the other hand, are usually discussed using three different "strengths": equivalences, isomorphisms, and adjunctions. This leads to the following question:

• Question: Have the "very lax" notions of $2$-dimensional co/limits corresponding to weakening the equivalences of categories in the above table to be merely adjunctions been studied before? Are there any interesting/natural examples of them?