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_D(X)⊗J)=y_C(X)⊗J is a subclass of projective acylic cofibrations (which are generated by y_C(X)⊗J for all X∈C), and every projective acyclic cofibration is an objectwise weak equivalence, which completes the proof.