Lemma 7.29.3. Let \mathcal{C}, \mathcal{D} be sites. Let u : \mathcal{C} \to \mathcal{D} be a special cocontinuous functor. For every object U of \mathcal{C} we have a commutative diagram
\xymatrix{ \mathcal{C}/U \ar[r]_{j_ U} \ar[d] & \mathcal{C} \ar[d]^ u \\ \mathcal{D}/u(U) \ar[r]^-{j_{u(U)}} & \mathcal{D} }
as in Lemma 7.28.4. The left vertical arrow is a special cocontinuous functor. Hence in the commutative diagram of topoi
\xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r]_{j_ U} \ar[d] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ u \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/u(U)) \ar[r]^-{j_{u(U)}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) }
the vertical arrows are equivalences.
Proof.
We have seen the existence and commutativity of the diagrams in Lemma 7.28.4. We have to check hypotheses (1) – (5) of Lemma 7.29.1 for the induced functor u : \mathcal{C}/U \to \mathcal{D}/u(U). This is completely mechanical.
Property (1). This is Lemma 7.28.4.
Property (2). Let \{ U_ i'/U \to U'/U\} _{i \in I} be a covering of U'/U in \mathcal{C}/U. Because u is continuous we see that \{ u(U_ i')/u(U) \to u(U')/u(U)\} _{i \in I} is a covering of u(U')/u(U) in \mathcal{D}/u(U). Hence (2) holds for u : \mathcal{C}/U \to \mathcal{D}/u(U).
Property (3). Let a, b : U''/U \to U'/U in \mathcal{C}/U be morphisms such that u(a) = u(b) in \mathcal{D}/u(U). Because u satisfies (3) we see there exists a covering \{ f_ i : U''_ i \to U''\} in \mathcal{C} such that a \circ f_ i = b \circ f_ i. This gives a covering \{ f_ i : U''_ i/U \to U''/U\} in \mathcal{C}/U such that a \circ f_ i = b \circ f_ i. Hence (3) holds for u : \mathcal{C}/U \to \mathcal{D}/u(U).
Property (4). Let U''/U, U'/U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}/U) and a morphism c : u(U'')/u(U) \to u(U')/u(U) in \mathcal{D}/u(U) be given. Because u satisfies property (4) there exists a covering \{ f_ i : U_ i'' \to U''\} in \mathcal{C} and morphisms c_ i : U_ i'' \to U' such that u(c_ i) = c \circ u(f_ i). We think of U_ i'' as an object over U via the composition U_ i'' \to U'' \to U. It may not be true that c_ i is a morphism over U! But since u(c_ i) is a morphism over u(U) we may apply property (3) for u and find coverings \{ f_{ik} : U''_{ik} \to U''_ i\} such that c_{ik} = c_ i \circ f_{ik} : U''_{ik} \to U' are morphisms over U. Hence \{ f_ i \circ f_{ik} : U''_{ik}/U \to U''/U\} is a covering in \mathcal{C}/U such that u(c_{ik}) = c \circ u(f_{ik}). Hence (4) holds for u : \mathcal{C}/U \to \mathcal{D}/u(U).
Property (5). Let h : V \to u(U) be an object of \mathcal{D}/u(U). Because u satisfies property (5) there exists a covering \{ c_ i : u(U_ i) \to V\} in \mathcal{D}. By property (4) we can find coverings \{ f_{ij} : U_{ij} \to U_ i\} and morphisms c_{ij} : U_{ij} \to U such that u(c_{ij}) = h \circ c_ i \circ u(f_{ij}). Hence \{ u(U_{ij})/u(U) \to V/u(U)\} is a covering in \mathcal{D}/u(U) of the desired shape and we conclude that (5) holds for u : \mathcal{C}/U \to \mathcal{D}/u(U).
\square
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