We prove that (a) if R is a left coherent ring, then the weak global dimension w D(R) <= n (n >= 2) if and only if every (n – 2)th F–cosyzygy of a finitely presented right R–module has a flat envelope with the unique mapping property; (b) if R is a left coherent and right perfect ring, then the right global dimension rD(R) <= n (n >= 2) if and only if every (n – 2)th P–cosyzygy of a right R–module has a projective envelope with the unique mapping property; (c) if R is a commutative ring, then R is π—coherent (resp. coherent) and the exactness of 0 -> K -> F₀ -> F₁ with F_o and F₁ (finitely) projective and K finitely generated implies the projectivity of K if and only if every finitely generated (resp, finitely presented) R–module has a (finitely) projective envelope with the unique mapping property.