A semiflexible limit is a strict 2-limit whose weight is semiflexible (defined below). The semiflexible weights are precisely those weights whose limits are bilimits.

(as a wide subcategory) has a strict left adjoint$Q$, which is the pseudo morphism classifier? for an appropriate strict 2-monad. Therefore, for any functor $\Phi \colon D\to Cat$, we have $Q\Phi \colon D\to Cat$ such that pseudonatural transformations $\Phi \to \Psi$ are in natural bijection with strict 2-natural transformations $Q\Phi \to \Psi$.

The counit of this adjunction is a canonical strict 2-natural transformation $q\colon Q\Phi \to \Phi$. We say that $\Phi$ is semiflexible if this transformation admits a pseudo-section? (equivalent is an equivalence) in $[D,Cat]$.