Astandard extension(resp.standard completion) is a function Z assigning to each poset P a (closure) system ZP of subsets such that x ⋚ y iff x belongs to every Z ε ZP with y ε Z. A poset P is Z -completeif each Z ε 2P has a join in P. A map f: P → P′ is Z—continuousif f−1[Z′] ε ZP for all Z′ ε ZP′, and a Z—morphismif, in addition, for all Z ε ZP there is a least Z′ ε ZP′ with f[Z] ⊆ Z′. The standard extension Z iscompositiveif every map f: P → P′ with {x ε P: f(x) ⋚ y′} ε ZP for all y′ ε P′ is Z -continuous. We show that any compositive standard extension Z is the object part of a reflector from IPZ, the category of posets and Z -morphisms, to IRZ, the category of Z -complete posets and residuated maps. In case of a standard completion Z, every Z -continuous map is a Z -morphism, and IR2 is simply the category of complete lattices and join—preserving maps. Defining in a suitable way so-calledZ -embeddingsand morphisms between them, we obtain for arbitrary standard extensions Z an adjunction between IPZand the category of Z -embeddings. Many related adjunctions, equivalences and dualities are studied and compared with each other. Suitable specializations of the function 2 provide a broad spectrum of old and new applications.