A lax limit is defined to be a 2-limit, except that the cone is only required to commute up to specified transformations, not up to isomorphism. In particular, the limit is defined up to isomorphism, and on examples such as product where there are no 2-cells in the cone, lax limits and 2-limits coincide.
I'm working with an example, whose properties are similar to smash product on partial orders with bottom. This gives rise to a lax version of a product, where the equalities $\langle f,g\rangle;p=f$ and $\langle f,g\rangle;q=g$ are replaced by 2-cells, and the uniqueness property becomes a universality property: for any other candidate $h$, we have a unique 2-cell $h\Rightarrow\langle f,g\rangle$.
Generalizing, it seems that there should be an "even laxer" notion of limit, where the adjunction used to define the limit is specified itself by an adjunction rather than by an equivalence of categories.
Does such a structure exist in the literature?