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252 U. Bunke, T. Schick, M. Spitzweck, and A. Thomfamily0 jY WD B@U i Y1 UC Ai2I:3.3.2 There is a canonical functor f W S=Y ! S given on objects by f.U ! Y/WD U .It induces adjoint pairs of functorsf W ShS=Y , ShS W f ;p f W Pr S=Y , Pr S W p f :We will often write f .F / DW F jY for the restriction functor. For F 2 Pr S it is givenin explicit terms byF jY .U ! Y/WD F.U/:The functor f is in fact the restriction of p f to the subcategory of sheaves ShS Pr S. IfF 2 ShS, then one must check that p f F is a sheaf. To this end note that thedescent conditions for p f F with respect to the induced coverings described in 3.3.1immediately follow from the descent conditions for F .3.3.3 Let us now study how the restriction acts on representable sheaves. We assumethat S has finite products. Let X; Y 2 S.Lemma 3.6. If S has finite products, then we have a natural isomorphismX jY Š X Y ! Y :Proof. By the universal property of the product for W U ! Y we haveX Y ! Y .U ! Y/D¹h 2 Hom S .U; X Y/j pr Y ı h D ºŠ Hom S .U; X/D X.U /D X jY .U ! Y/:3.3.4 Let i ] W Pr S ! ShS and i ] YW Pr S=Y ! ShS=Y be the sheafification functors.Lemma 3.7. We have a canonical isomorphism f ı i ] Š i ] Y ı p f .Proof. The argument is similar to that in the proof of [BSS, Lemma 2.31]. The mainpoint is that the sheafifications i ] F.U/ and i ] Y F jY .U ! Y/ can be expressed interms of the category of covering families cov S .U / and cov S=Y .U ! Y/. Comparedwith [BSS, Lemma 2.31] the argument is simplified by the fact that the induction ofcovering families described in 3.3.1 induces an isomorphism of categories cov S .U / !cov S=Y .U ! Y/.