Collectionwise weak continuity duals

A function f: (X, τ) → (Y, σ) is weakly collectionwise continuous if for some C ⊆ 2X with τ ⊆ C we have f−1(V) ∈ C for each V ∈ σ. In this case, f is said to be C-continuous. If also τ ⊆ C* ⊆ 2X, C*-continuity is a dual to C-continuity if C⋂C* = τ and then the pair (C-continuity, C*-continuity) is a decomposition of continuity. In this paper, two natural topological methods are found for construction of a dual to any collectionwise weak continuity. Some known decompositions are improved.