Coherence for bicategorical cartesian closed structure

Abstract We prove a strictification theorem for cartesian closed bicategories. First, we adapt Power’s proof of coherence bicategories with finite bilimits to show that every bicategory bicategorical structure is biequivalent 2-category 2-categorical structure. Then how extend this result Mac Lane-style “all pasting diagrams commute” theorem: precisely, in the free on graph, there at most one 2-cell between any parallel pair 1-cells. The argument employ reminiscent used by Čubrić, Dybjer, and Scott normalisation simply-typed lambda calculus (Čubrić et al., 1998). main results first appeared conference paper (Fiore Saville, 2020) but reasons space many details are omitted there; here provide full development.