  We introduce an operad which acts on the Gerstenhaber-Schack complex of a
prestack as defined by Dinh Van and Lowen, and which in particular allows us to
endow this complex with an underlying $L_{\infty}$-structure. We make use of
the operad $\operatorname{Quilt}$ which was used by Hawkins in order to solve
the presheaf case. Due to the additional difficulty posed by the presence of
twists, we have to use $\operatorname{Quilt}$ in a fundamentally different way
(even for presheaves) in order to allow for an extension to prestacks. The
resulting $L_{\infty}$-algebra governs the deformation theory of the prestack.
