Definition 7.1. Suppose C and D are two framed sites. A functor F: C ® D is continuous if all the objects and morphisms in C are geometric (therefore strongly geometric) over D (i.e., C = Geo(C/F)). 

Remark 7.2. (a) Any bicontinuous functor of framed sites is continuous. 
(b) A composition of two continuous functors is continuous. 
(c) Suppose G: E ® C is a bicontinuous functor of framed sites and F: C ® D is any functor. Then FG: E ® D is continuous if and only if G(E) Í Geo(C/F). 
(d) The restriction of any functor F: C ® D on the subsite Geo(C/F) of C is a continuous functor, denoted also by F. 

Proposition 7.3. Suppose F: C ® D is a continuous functor. Suppose X is an object of C and W is an open  sieve on F(X). Then F_X^-1(W) Í F_X^*(W). If F is strict and W is open effective then  F_X^*(W) = F_X^-1(W). 

Proof. Suppose f: Y ® X is a morphism in F_X^-1(W). Then F(f): F(Y) ® F(X) is in W, so 1_D/F(Y) Í F(f)^-1(W). Thus 1_C/Y = F_Y^*(1_D/F(Y)) Í F_Y^*(F(f)^-1(W)) = f^-1(F_X^*(W)), which implies that f is in F_X^*(W). Thus F_X^-1(W) Í F_X^*(W). If F is strict and W is open effective then F_X^*(W) Í F_X^-1(W) by (6.2.d), hence F_X^*(W) = F_X^-1(W). n 

Theorem 7.4. Suppose F: C ® D is a strict functor. Then F is continuous if and only if for any object X of C and any open effective cover {W_i} of F(X), {F_X^-1(W)} is an open cover of X. 

Proof. One direction comes from (7.3). For the other direction suppose the condition holds for F. Since any open effective sieve W on F(X) can be included in an open effective cover of F(X), F_X^-1(W) is open, and so F_X^*(W) = F_X^-1(W). It follows that X is geometric over D, and each morphism f: Y ® X is geometric because f^-1(F_X^-1(W)) = F_Y^-1(F(f)^-1(W)) holds (unconditionally for any sieve W). n 

An object X of a framed site C is called local if the joint M(X) of all the open sieves U ¹ 1_C/X of X is not 1_C/X. A morphism f: Y ® X of local objects of C is called a local morphism if f^-1(M(X)) ¹ 1_C/Y. 

Remark 7.5. The following conditions for an object X of C are equivalent: 
(a) X is a local object. 
(b) G(X) is a local locale (i.e., a local object in Loc). 
(c) Any open cover of X contains 1_C/X. 

Remark 7.6. Suppose X, Y and Z are local objects of C and f: Y ® X, g: Z ® Y are morphisms in C. 
(a) If f and g are local morphisms, then fg is a local morphism. 
(b) If fg is a local morphism then f is a local morphism. 
 
Proposition 7.7. Suppose F: C ® D is a continuous functor of framed sites. 
(a) If X is a local object of C, then F(X) is a local object of D and G(F_X): G(X) ® G(F(X)) is a local morphism of local locales. 
(b) If X and Y are two local objects of C and f: Y ® X is a local morphism, then F(f): F(X) ® F(Y) is a local morphism of local objects of D. 

Proof. (a) First we show that if U is an open sieve on F(X) and F_X^*(U) = 1_X, then U = 1_F(X). The assumption  F_X^*(U) = 1_X implies that there is an open cover {V_i} of X such that each restriction F(V_j) ® F(X) is in U. Since X is local there is some i such that V_i = 1_X by (7.5.c). Thus F(X) ® F(X) is in U, hence U = 1_F(X). 
Now suppose {U_i} is an open cover of F(X). Since F is continuous, F_X^*: G(F(X)) ® G(X) is a morphism of frames by (6.6.a), thus {F_X^*(U_i)} is an open cover of X. Since X is a local object, there exits some i such that F_X^*(U_i) = 1_X by (7.5.c). It follows from the above observation that U_i = 1_F(X). We have proved that any open cover of F(X) contains 1_X, thus F(X) is local by (7.5.c). Since M(F(X)) ¹ 1_F(X), the above assertion also implies that F_X^*(M(F(X)) ¹ 1_X. Thus G(F_X): G(X) ® G(F(X)) is a local morphism of local locales. 
(b) Since F is continuous we have G(F_X)G(f) = G(F(f))G(F_Y). As G(F_X) and G(f) are local morphisms, G(F(f)) is a local morphism by (7.6). n