"Very lax" $2$-dimensional co/limits

Question

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:

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?

Answer 1

I believe these kinds of (co)limits are the ones defined in Gray's Formal Category Theory: Adjointness for $2$-categories I.7.9.1 (as quasi-(co)limits); some examples and computations of them are given further on in I.7.10, I.7.11 and I.7.12, and recover several classical constructions such as (co)comma objects or Eilenberg–Moore (or Kleisli) categories, as well as the usual description, when the target $2$-category is $\mathfrak{Cat}$, in terms of sections of an associated opfibration. Note that Gray defines them in the $3$-dimensional—or rather, (lax) Gray-categorical—setting, using his notion of quasi-adjunction, a.k.a. soft adjunction or lax adjunction or local adjunction ("adjunction up to adjunctions"), this last description (established in MacDonald–Stone's Soft adjunction between $2$-categories) being what recovers the definition you are asking for.

However, I have not read this book in enough detail to be able to answer confidently on what exactly Gray's somewhat idiosyncratic terminology corresponds to (I am only writing this as an answer because it is slightly too long for a comment), though various flavours of laxity, not all of which have stood the test of time, are allowed—see the discussion at the end of Definition I.7.1 for a list; I think the relevant kind for your question (and in general use) should be "strict weak quasi-adjunction" or "transcendental quasi-adjunction".