Definition Weighted limit [efr-L9JB]

Let \mathbb {C} be a 2-category, let D: \mathbb {I} \to \mathbb {C} be a 2-diagram in it, and let W: \mathbb {I} \to \mathsf {Cat} be another 2-functor. A limit of D weighted by {W} is an object \lim ^W D \in \mathbb {C} equipped with a natural isomorphism of categories \mathbb {C}(X,\lim ^W D) \cong [ {\mathbb {I} }, {\mathbb {C}} ] (W, \mathbb {C}(X,D(-))) (natural in X \in \mathbb {C}).