  A crossed module constitutes a strict $2$-groupoid $\mathcal{G}$ and a
$\mathcal{G}$-valued cocycle on a manifold defines a $2$-bundle. A
$2$-connection on this $2$-bundle is given by a Lie algebra $\mathfrak g$
valued $1$-form $A $ and a Lie algebra $\mathfrak h$ valued $2$-form $B $ over
each coordinate chart together with $2$-gauge transformations between them,
which satisfy the compatibility condition. Locally, the path-ordered integral
of $A $ gives us the local $1$-holonomy, and the surface-ordered integral of
$(A ,B )$ gives us the local $2$-holonomy. The transformation of local
$2$-holonomies from one coordinate chart to another is provided by the
transition $2$-arrow, which is constructed from a $2$-gauge transformation. We
can use the transition $2$-arrows and the $2$-arrows provided by the
$\mathcal{G}$-valued cocycle to glue such local $2$-holonomies together to get
a global one, which is well defined. Namely we give an explicit algorithm for
calculating the global $2$-holonomy.
