Let A be a Banach algebra and X be a compact Hausdorff space.For given homomorphisms σ ∈ Hom(A) and τ ∈ Hom(C(X, A)), we introduce homomorphismsσ ∈ Hom (C(X, A)) andτ x ∈ Hom(A), where x ∈ X. We then study bothσ-(weak) amenability of C(X, A), andτ x -(weak) amenability of A.