  We define the notion of adjustment for strict Lie 2-groups and provide the
complete cocycle description for non-Abelian gerbes with connections whose
structure 2-group is an adjusted 2-group. Most importantly, we depart from the
common fake-flat connections and employ adjusted connections. This is an
important generalisation that is needed for physical applications especially in
the context of supergravity. We give a number of explicit examples; in
particular, we lift the spin structure on $S^4$, corresponding to an
instanton-anti-instanton pair, to a string structure, a 2-group bundle with
connection. We also outline how categorified forms of Bogomolny monopoles known
as self-dual strings can be obtained via a Penrose-Ward transform of string
bundles over twistor space.
