Now ϕ_q−1 ◦ ∂_q = ∂_q ◦ ϕ_q for each integer q. Therefore ϕ_q(Z_q(K)) ⊂ Z_q(L)  and ϕ_q(B_q(K)) ⊂ B_q(L) for all integers q. It follows that any simplicial map ϕ: K → L induces well-defined homomorphisms ϕ∗: H_q(K) → H_q(L) of homology groups, where ϕ∗([z]) = [ϕ_q(z)] for all q-cycles z ∈ Z_q(K). It is a trivial exercise to verify that if K, L and M are simplicial complexes and if ϕ: K → L and ψ: L → M are simplicial maps then the induced homomorphisms of homology groups satisfy (ψ ◦ ϕ)_∗ = ψ_∗ ◦ ϕ_∗.