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

[![enter image description here][1]][1]

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

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

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?


  [1]: https://i.sstatic.net/QtYdC.jpg