Let I: D ® C be a covariant functor of categories. Suppose X is an object of C. An object J(X) of D together with a morphism r: I(J(X)) ® X (resp. r: X ® I(J(X))) is called a left (resp. right) associated object of X in D if it has the following universal property: 
For any morphism f: I(Z) ® X (resp. f: X ® I(Z)) in C with Z Î D, there exists a unique morphism g: Z ® J(X) (resp. g: J(X) ® Z) such that f = rI(g) (resp. f = I(g)r). 

We say that (J(X), r) is a strict left (resp. right) associated object of X if I(J(X)) = X and r is the identity morphism 1_X. 

An associated object of X is uniquely determined by X and I up to isomorphism. The morphism r is called the counit (resp. unit) of X. If Y Î D and I(Y) has a left (resp. right) associated object J(I(Y)) in D, then the unique morphism s: Y Î J(I(Y)) (resp. s: J(I(Y)) ® Y) such that 1_I(Y) = rI(s) (resp. 1_I(Y) = I(s)r) is called the unit (resp. counit) of Y. 

Clearly I: D ® C has a right (resp. left) adjoint J: C ® D if and only if any object X of C has a left (resp. right) associated object J(X). 

Recall that if D is a subcategory of C and the inclusion functor I: D ® C has a right (resp. left) adjoint, then D is called a coreflective (resp. reflective) subcategory of C. 

Example 4.1.1. Let D be a coreflective (resp. reflective) subcategory of C. Then any object X of D together with the identity morphism 1_X: X ® X is a left (resp. right) associated object of X if and only if D is a full subcategory of C. 

Example 4.1.2. Suppose D is a coreflective subcategory of a category C, and D' a coreflective subcategory of D. Then D' is a coreflective subcategory of C. 

Definition 4.1.3. Suppose C is an everywhere effective site. An everywhere effective subsite D of C is called embedded if D is a coreflective subcategory of C such that for any object X e D, the unit s: X ® J(X) is an effective morphism in D. 

Remark 4.1.4. If D is a full coreflective subsite of C, then J(X) = X for any X Î D and s: X ® X is the identity morphism 1_X of X (4.1.1). Thus D is embedded. 

Proposition 4.1.5. Suppose C is an everywhere effective site, D an everywhere effective coreflective subsite of C having the following properties: 
(a) For any object X Î D and any subset U of |X|, the inclusion U ® X (in C) is in D. 
(b) Suppose X, Y are objects of D. Then f: X ® Y is in D if and only if for any x Î |X|, the morphism f_x: x ® f(x) of effective points is a morphism of D. 
(c) Suppose f: X ® Y, g: Y ® Z are morphisms of C and f is surjective. Suppose f and gf are in D. Then g is in D. 
Then D is an embedded subsite of C. 

Proof. Suppose X ® D and Y = J(X) is a left associated object of X in D with the unit s: X ® Y and counit r: Y ® X. Then we have 
(d) r.s = 1_X. 
(e) If h, h': Z ® Y are any two morphisms in D and rh = rh', then h = h'. We shall prove the following three facts: 
(i) For any morphism h: Z ® Y in D with |h(Z)| Í |s(X)| we have rh Î D. 
(ii) s: X ® Y is effective in D. 
(iii) |s(X)| = {y Î |Y| ½ r_y: y ® r(y) is in D}. 
For any x Î |X| we have (rs)_x = r_s(x)s_x = 1_x Î D and s_x: x ® s(x) is in D by (b). It follows that r_s(x) Î D by (c). Suppose h: Z ® Y is a morphism in D with |h(Z)| Í |s(X)|. For any z Î |Z| we have h(z) Î |s(X)|, thus r_h(z) Î D. Since h Î D, h_z Î D by (b). Thus (rh)_z = r_h(z)h_z Î D for any z Î |Z|. This means that rh Î D by (b), hence (i) holds. 
The morphism s: X ® Y is a bicontinuous monomorphism by (d). Suppose h: Z ® Y is a morphism in D and |h(Z)| Í |s(X)|. We have rh Î D by (i), hence srh Î D. Then rsrh = 1_Xrh = rh implies that srh = h by (e). This proves that h factors through s by the morphism rh Î D (the uniqueness of rh is obvious since s is monomorphic). This proves (ii). 
Suppose y Î |Y| and i: y ® Y is the inclusion morphism in D. If y Î |s(X)| then r_y = ri: y ® X is in D by (i). If y Ï |s(X)| then we have sri(y) ¹ i(y) but rsri(y) = ri(y) as rs = 1_X. Since i Î D we must have sri Î D by (e), which implies that r_y = ri Î D because s e D. Thus (iii) holds. 

Proposition 4.1.6. (Chevalley, cf. [H] p.68) Any ringed space (X, O) has a left associated object in the category LSp of local ringed spaces. The subsite LSp is an embedded subsite of the site RSp of ringed spaces. 

Proof. Let Spec X = {(x, z)|x Î X and z is a prime ideal of O_x}. For any (x, z) Î Spec X we shall write A_z for the localization of O_x at z. We take the topology on Spec X generated by the open subsets U_g = <U, g>, where U is an open subset of X and g Î O(U), such that <U, g> = {(x, z)|x Î U and g_x Ï z}. For any open subset W of Spec X we define O_SpecX(W) to be the ring of functions s: W ® È_(x,z)ÎW A_z such that s(x, z) Î A_z for each (x, z), and such that for any (x, z) Î W, there is a small neighborhood (V, g) of (x, z) contained in W and an elements a of O(V) such that and s(x', z') = a/g in A_z' for any (x', z') Î (V, g). We thus defined a sheaf O_SpecX on Spec X. For each point (x, z) Î Spec X it is easy to see that the stalk of O_SpecX at (x, z) is isomorphic to A_z. Thus (Spec X, O_SpecX) is a local ringed space. Let j: Spec X ® X be the natural map sending any (x, z) Î Spec X to x Î X. The natural homomorphism O(U) ® O_SpecX(j^-1(U)) induces a morphism of ringed spaces (j, j^#) from Spec X to X. 
One can easily prove that (Spec X, O_SpecX) together with the morphism (j, j^#) has the universal property required. This proves that LSp is a coreflective subsite of RSp. 
We verify the conditions of (4.1.5.) for LSp. First (4.1.5.a & b) follow directly from the definition of morphisms of local ringed spaces. For (4.1.5c) note that if f^#: O_y ® O_x, g^#: O_z ® O_y are morphisms of local rings, and if f^# and f^#g^# are local, then g^# is local. This proves that LSp is an embedded subsite of RSp. 

Proposition 4.1.7. GSp is an embedded subsite of LSp. 

Proof. For any local ringed space X the geometric space X_red = (X, O_X/I) is a left associated object of X in the category of geometric spaces. We obtain a right adjoint functor red: LSp ® GSp. Gsp is a full subsite of LSp, hence an embedded subsite. 

Proposition 4.1.8. RSet is an embedded subsite of RSp. LSet and GSet are embedded subsites of LSp. GSet is an embedded subsite of GSp. 

Proof. Any ringed space X determines a ringed set f(X) = {O_x|x Î X} canonically. If X is a local ringed space, then f(X) is a local ringed set, and f_k(X) = {k_x|x Î X} is a geometric set, where k_x is the residue field of O_x. Clearly f: RSp ® RSet, f: LSp ® LSet, f_k: LSp ® GSet and f_k: GSp ® GSet are right adjoint functor of the inclusion functors. Also these subsites are full, hence are embedded.