[For part I, see A. Carboni, G. M. Kelly and R. J. Wood, Cah. Topologie Géom. Différ. Catégoriques 32, No. 1, 47-95 (1991; Zbl 0747.18008).]
For these authors, equipment for a category \({\mathcal K}\) is essentially a category \({\mathcal E}\) together with a parallel pair of functors \({\mathcal E}\rightrightarrows{\mathcal K}\) which form a 2-sided fibration \({\mathcal K}\leftarrow{\mathcal E}\to {\mathcal K}\) from \({\mathcal K}\) to \({\mathcal K}\) in the sense of the reviewer [R. Street, “Fibrations and Yoneda’s lemma in a 2-category”, Lect. Notes Math. 420, 104-133 (1974; Zbl 0327.18006)]. The 2-category EQT of equipments has the arrows and 2-cells appropriate to a diagram 2-category of parallel pairs. Adjunctions in EQT are examined. This is applied to study aspects of the 2-functor spn: PBK\(\to\)EQT, where PBK is the full sub-2-category of CAT consisting of categories admitting pullbacks, and spn\(({\mathcal K})\) is the equipment for \({\mathcal K}\) whose objects are spans in \({\mathcal K}\).