Let ( M , ∂ M ) (M,\partial M) be a compact n n -manifold with boundary, orientable over a field K K with characteristic q q . For f : ( Y , ∂ Y ) → ( M , ∂ M ) f:(Y,\partial Y) \to (M,\partial M) , with Y Y compact, and ( X , ∂ X ) (X,\partial X) a compact pair, g : X → M g:X \to M , let ( P , ∂ P ) = { ( y , x ) ∈ Y × ( X , ∂ X ) | f ( y ) = g ( x ) } (P,\partial P) = \{ (y,x) \in Y \times (X,\partial X)|f(y) = g(x)\} denote the fibered product, with p p as the projection to ( X , ∂ X ) (X,\partial X) . In Čech-cohomology with coefficients K K , we show that if [ u n k ] n ( f ) {[unk]^n}(f) is injective then so is [ u n k ] ∗ ( p ) {[unk]^*}(p) —and a number of strengthenings, which point to a concept of q q -essential map from one compact space to another.