Given an ∞-bicategory D with underlying ∞-category D_0, we construct a Cartesian fibration Tw(D) --> Tw(D_0) x D_0^op, which we call the enhanced twisted arrow ∞-category, classifying the restricted mapping category functor Map_D:D_0^op x D_0 --> D^op \times D --> Cat_∞. With the aid of this new construction, we provide a description of the ∞-category of natural transformations Nat(F,G) as an end for any functors F and G from an ∞-category to an ∞-bicategory. As an application of our results, we demonstrate that the definition of weighted colimits studied by Gepner-Haugseng-Nikolaus satisfies the expected 2-dimensional universal property.
Keywords: (∞,2)-category, twisted arrow category, natural transformation, weighted colimit
2020 MSC: 18N10, 18N60, 18N65
Theory and Applications of Categories, Vol. 39, 2023, No. 4, pp 98-149.