Let f : X → Y be a perfect surjective map of metrizable spaces. It is shown that if Y is a C-space (resp., dimY ≤ n and dim f ≤ m), then the function space C(X, I ∞ ) (resp., C(X, I 2n+1+m )) equipped with the source limitation topology contains a dense Gδ-set H such that f ×g embeds X into Y × I ∞ (resp., into Y × I 2n+1+m ) for every g ∈ H. Some applications of this result are also given.