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.

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.