Coherence for adjunctions in a 3-category via string diagrams

Abstract

We construct a 3-categorical presentation Adj(3,1) and define a coherent adjunction in a strict 3-category C as a map Adj(3,1)→C. We use string diagrams to show that any adjunction in C can be extended to a coherent adjunction in an essentially unique way. The results and their proofs will apply in the context of Gray 3-categories after the string diagram calculus is shown to hold in that context in an upcoming paper.