A flexible limit is a strict 2-limit whose weight is cofibrant. This implies that flexible limits are also 2-limits (in the non-strict sense, which for us is the default – recall that these are traditionally called bilimits).
Furthermore, all PIE-limits and therefore all strict pseudo-limits are flexible; thus any strict 2-category which admits all flexible limits also admits all -limits. A number of strict 2-categories admit all flexible limits, but not all strict -limits, and this is a convenient way to show that they admit all -limits.
Let be a small strict 2-category. Write for the strict 2-category of strict 2-functors, strict 2-natural transformations?, and modifications, and for the strict 2-category of strict 2-functors, pseudonatural transformations, and modifications. The inclusion
(as a wide subcategory) has a strict left adjoint , which is the pseudo morphism classifier? for an appropriate strict 2-monad. Therefore, for any functor , we have such that pseudonatural transformations are in natural bijection with strict 2-natural transformations .
All PIE-limits are flexible. This includes products, inserters, equifiers by definition, and also descent objects, iso-inserters, comma objects, and so on. In fact, PIE-limits have a characterization similar to the definition above of flexible limits: they are the coalgebras for regarded as a 2-comonad.
The splitting of idempotents is flexible, but not PIE. Moreover, in a certain sense it is the “only” such. Precisely, flexible limits are the saturation of each of the following classes of limits: * PIE-limits together with splitting of idempotents * PIE-limits together with splitting of idempotent equivalences * strict pseudo-limits together with splitting of idempotents * strict pseudo-limits together with splitting of idempotent equivalences