I use the notation and 2-categorical conventions from the papers by Lack, Kelly, Power, et. al. about 2-monad theory and 2-limits. When $K$ is a 2-category which is complete and has copower, then one may express the weighted limit functor $\operatorname {lim}_W$ has a $Cat$-enriched right adjoint of the 2-functor $\Delta^W:K \to K^\mathscr J$ which sends some $A$ to the diagram $\Delta_W A = W(-)\ast A$. Here $\ast$ denotes the copower 2-functor. This is explained here.
Given a fixed weight $W$, the pseudolimit can be turned into a 2-functor $\operatorname{pslim}_W:\operatorname{Ps}(\mathscr J,K)\to K$. Is the pseudo-limit also a $Cat$-enriched right adjoint? What about bilimits? I suspect that one can turn them into right-biadjoints by using a similar trick, but with the weaker notion of a bicopower. Is that correct?