We can apply the Kan transfer theorem (Theorem 11.3.2 in Hirschhorn)
to the right adjoint functor U: M^C→M^D, where M^D is equipped with the projective model structure.
Its left adjoint F sends representable functors y(X) on D
to the corresponding representable functors y(X) on C.
The conditions in the theorem boil down to verifying that U takes relative
F(y(X)⊗J)-complexes to weak equivalences (J denotes a set of generating
acyclic cofibrations of M), where X∈D.

However, F(y(X)⊗J) is a subclass of projective acylic cofibrations,
and every projective acyclic cofibration is an objectwise weak equivalence,
which completes the proof.