  Building on the work of Gerstenhaber and Schack for presheaves of algebras,
we define a Gerstenhaber-Schack complex C_GS(A) for an arbitrary prestack A,
that is a pseudofunctor taking values in linear categories over a commutative
ground ring. In the general case, the differential is no longer simply the sum
of Hochschild and simplicial contributions as in the presheaf case, but
contains additional higher components as well. If A' denotes the Grothendieck
construction of A, which is a map-graded category, we explicitly construct
inverse quasi-isomorphisms between C_GS(A) and the Hochschild complex C(A'). As
the Homotopy Transfer Theorem applies to our construction, one can transfer the
dg Lie structure present on the Hochschild complex in order to obtain an
L_infinity structure on C_GS(A), which controlls the higher deformation theory
of the prestack A.
