Theorem 7.1 ([Quillen equivalence]). The adjunction L_n:_KQ (,C_n,W_n,F_n):R_n is a Quillen equivalence.

Equations and show that R_n preserves fibrations and trivial fibrations, so this is a Quillen adjunction. Set S= and T= , and let a_K:K→TSK be the unit of S T . The mate identity is j_K=T(ℓ_K)a_K:K⟶TK. The independently cited theorem above makes j_K a weak equivalence. Since ℓ_K is a weak equivalence and T preserves weak equivalences, T(ℓ_K):TSK→TK is also one. Hence a_K is a weak equivalence for every simplicial set K , by two-out-of-three. In particular, a_ SK :SK→TS^2K is a weak equivalence. The unit of S^2 T^2 is b_K=T(a_ SK )a_K:K⟶T^2S^2K, T(a_ SK ):TSK⟶T^2S^2K. Thus b_K is a weak equivalence because T preserves weak equivalences. Composition is written with the rightmost map applied first. The unit of the composite adjunction L_n R_n is u_K=T^2(θ_K)b_K: K⟶T^2 c_nS^2K=R_nL_nK. Proposition 22 ([The unit after two subdivisions]), preservation of weak equivalences by T , and the preceding calculation show that u_K is a weak equivalence for every K . For any strict functor f:L_nK→Y , its simplicial adjunct is f^=R_n(f)u_K:K⟶R_nY. Using the definition of W_n and two-out-of-three now gives f∈W_n R_n(f)∈W_ KQ f^∈W_ KQ . This equivalence holds for every K and Y . In particular it holds for cofibrant K and fibrant Y , which is the Quillen-equivalence criterion. Together with Propositions Proposition 20 ([The transferred model structure]) and Proposition 21 ([Properness]), this proves all the asserted model-categorical conclusions for each finite n≥1 and for n=∞ .