Suppose (C, G) is a framed site. If U is a sieve on an object X of C we denote by U* the joint of open subsieves U_i of U in G(X) (i.e. U* = Ú_i U_i). 

Proposition 6.1. Suppose U and V are two sieves on X. 
(a) Suppose V is open. Then V Í U* if and only if V has an open cover {V_i} with each V_i Í U. 
(b) (U Ç V)* = U* Ç V*. 
(c) If f: Y ® X is a morphism then f^-1(U*) Í (f^-1(U))*. 
(d) If f: Y ® X is an open effective morphism then f^-1(U*) = (f^-1(U))*. 
(e) If C is neat then U Í U*. 
( 
Proof. (a) If V Í U* then V = V Ç U* = V Ç (Ú_i U_i) = Ú_i V Ç U_i. This shows that {V_i = V Ç U_i | U_i is any open subsieve of U} is an open cover of V. The other direction is trivial. 
(b) Clearly we have (U Ç V)* Í U* Ç V*. Since U* Ç V* Í U*, U* Ç V* has an open cover {U_i | iÎ I} with each U_i Í U by (a). Similarly U* Ç V* has an open cover {V_j | j Î J} with each V_j Í V. Then {U_i Ç U_j | i Î I and j Î J} is an open cover of U* Ç V* and each U_i Ç V_j Í U Ç V. It follows that U* Ç V* Í (U Ç V)* by (a). The other direction is trivial. 
(c) Suppose {U_i} is an open cover of U* with each U_i Í U, then f^-1(U_i) is an open cover of f^-1(U*) and f^-1(U_i) Í f^-1(U). Thus f^-1(U*) Í (f^-1(U))* by (a). 
(d) Suppose Y = V for an open effective sieve V on X. Then f^-1(U*) = V Ç U* = V* Ç U* = (V Ç U)* =  (f^-1(U))*. 
(e) If C is neat any sieve U on X is the union of its open effective subsieves. By definition U* is the joint of open sieves on U, so U Í U*. 

Suppose (C, G_C) and (D, G_D) are two framed sites. We consider a fixed functor F: C ® D. Suppose X is an object of C. If W is any sieve on F(X) we write F_X*(W) for (F_X^-1(W))*. We obtain an order-preserving map F_X*: w_D(F(X)) ® G_C(X). 

Proposition 6.2. (a) If U and V are sieves on F(X), then F_X*(U Ç V) = F_X*(U) Ç F_X*(V). 
(b) For any morphism f: Y ® X in C and any sieve W on F(X) we have f^-1(F_X*(W)) Í F_Y*(F(f)^-1(W)). 
(c) For any open effective morphism f: Y ® X in C and any sieve W on F(X) we have f^-1(F_X*(W)) = F_Y*(F(f)^-1(W)). 
(d) Suppose  F: C ® D is a strict functor. Suppose X is an object of C and W an active sieve on F(X). Then F_X*(W) Í F_X^-1(W). If C is neat then F_X*(W) = F_X^-1(W). 
 
Proof. (a) We have F_X*(U Ç V) = (F_X^-1(U Ç V))* = (F_X^-1(U) Ç F_X^-1(V))* = (F_X^-1(U))* Ç (F_X^-1(V))* = F_X*(U) Ç F_X*(V) by (6.1.b). 
(b) We have f^-1(F_X*(W)) Í (f^-1(F_X^-1(W))* = (F_Y^-1(F(f)^-1(W))* = F_Y*(F(f)^-1(W)) by (6.1.c). 
(c) We have f^-1(F_X*(W)) = (f^-1(F_X^-1(W))* = (F_Y^-1(F(f)]^-1(W))* = F_Y*(F(f)^-1(W)) by (6.1.d). 
(d) Suppose {U_i} is an open cover of F_X*(W) such that each U_i is an open subsieve of F_X^-1(W). Suppose f: Y ® X is a morphism in F_X*(W). Then U = f^-1(È {U_i}) is the unoin of an open cover of Y. Since F is strict, F(Y) is a colimit of F(U) (i.e., F(Y) = F(U); cf. (1.3)). For each g Î U we have F(fg) Í W, thus F(fg) can be factored uniquely through the inclusion morphism W ® F(X). This implies that the morphism F(f): F(Y) ® F(X) can be factored uniquely through the inclusion W ® F(X), i.e., f Î F_X^-1(W). This shows that F_X*(W) Í F_X^-1(W). If C is neat then F_X^-1(W) Í (F_X^-1(W))* = F_X*(W) by (6.1.e), thus F_X*(W) = F_X^-1(W). 

Definition 6.3. We say an object X of C is geometric over D (for F) if F_X* maps an open cover of F(X) to an open cover of X. We say that X is strongly geometric over D if any open subobject of X is geometric over D. A morphism f: Y ® X in C is a geometric morphism over D if f^-1(F_X*(W)) = F_Y*(F(f)^-1(W)) for any open effective sieve W of F(X). We say that f is strongly geometric over D if for any open effective sieve U of X 
with V = f^-1(U), the restriction f|_V: V ® U is geometric. 

Proposition 6.4. (a) Any open subobject of a strongly geometric object over D is strongly geometric. 
(b) Any open effective morphism in C is strongly geometric over D. 
(c) A composition of strongly geometric morphisms of strongly geometric objects over D is a strongly geometric morphism. 

Proof. (a) and (c) follow directly from the definition (6.3), and (b) is the content of (6.2.c). 

It follows from (6.4) that the collection of strongly geometric objects with the strongly geometric morphisms over D forms a subsite Geo(C/F) of C, called the geometric subsite of C over D. 

Lemma 6.5. Suppose U is a geometric open subobject of an object X of C. Suppose U Í F_X^-1(W) for an open sieve W on F(X)  and {W_i} is an open cover of W. Then {U Ç F_X*(W_i)} is an open cover of U. 

Proof. Applying (6.2.c) to the open effective morphism f: U ® X we get U Ç F_X*(W_i) = f^-1(F_X*(W_i)) = F_U*(F(f)^-1(W_i)). Since {W_i} is an open cover of F(X), {F(f)^-1(W_i)} is an open cover of F(U). But U is geometric over C, so {F_U*(F(f)^-1(W_i))} is an open cover of U. This implies that {U Ç F_X*(W_i)} is an open cover of U. 

Proposition 6.6. (a) Suppose X is strongly geometric over D. The map F_X*: G_C(F(X)) ® G_D(X) is a morphism of frames (i.e., G(F_X): G_D(X) ® G_C(F(X)) is a morphism of locales). 
(b) Suppose X and Y are strongly geometric objects over D and f: Y ® X is a geometric morphism over D. Then f^-1(F_X*(W)) = F_Y*(F(f)^-1(W)) for any open sieve W of F(X) (i.e., G(F_X)G(f) = G(F(f))G(F_Y)). 

Proof. (a) We only need to show that F_X* preserves joints in view of (6.1.b). Suppose W is an open sieve on F(X) and {W_j | j Î J} is an open cover of W. Let {U_i | i Î I} be an open effective cover of F_X*(W) with U_i Í F_X^-1(W). Since X is strongly geometric, each U_i is geometric, thus by (6.5) {U_i Ç F_X*(W_j) | j Î J} is an open cover of U_i. This shows that {F_X*(W_j) | j Î J} is an open cover of F_X*(W). 
(b) Since X and Y are strongly geometric objects over D, F_X* and F_Y* are morphisms of frames by (a). Thus f^-1F_X* and F_Y*F(f)^-1 are two morphisms of frames from G(F(X)) to G(Y). Since f is geometric over D, these two maps agree at any open effective sieve on F(X), and therefore also agree at any open sieve on F(X) because open effective sieves on F(X) form a base for G(F(X)). 

Theorem 6.7. Suppose X is an object of C and {U_i | i Î I} an open effective cover of X. 
(a) If each U_i is geometric (resp. strongly geometric) over D, then X is geometric (resp. strongly geometric) over D. 
(b) Suppose X and Y are strongly geometric objects over D. Suppose f: Y ® X is a morphism such that the restriction f|_Vi: V_i ® U_i of f on each V_i = f^-1(U_i) is strongly geometric. Then f is strongly geometric over D. 

Proof. (a) First suppose each U_i is geometric over D. Suppose {W_j | j Î J} is an open cover of F(X). Then by (6.5) {U_i Ç F_X*(W_j) | j Î J} is an open cover of U_i. Thus {F_X*(W_j) | j Î J} is an open cover of X, i.e., X is geometric over D. 
Next suppose each U_i is strongly geometric. Then for any open subobject U of X there exist an open effective cover of U consisting of geometric objects over C. Applying the above result we see that U is geometric. Thus X is strongly geometric over D. 
(b) Under the assumption we have G(F_Ui)G(f|_Vi) = G(F(f|_Vi))G(F_Vi) for each i by (6.6.b). Since G(F_Ui) and G(F(f|_Vi)) are morphisms of locales by (6.6.a), G(F_Ui)G(f|_Vi) and G(F(f|_Vi))G(F_Vi) are morphisms of locales. This implies that the restrictions of the morphisms of locales G(F_X)G(f) and G(F(f))G(F_Y) on each V_i are equal. Since {V_i} is an open effective cover of Y, we have G(F_X)G(f) = G(F(f))G(F_Y) as Loc is strict by (5.1). This proves that f is geometric. For any open effective sieve U of X with V = f^-1(U), the induced morphism f|_V: V ® U satisfies the same condition, thus f|_V is geometric, hence f is strongly geometric over D.