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260 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3.3.16 We now consider the derived version of Lemma 3.20. Let F 2 Sh Ab S andG 2 Sh Ab S 0 be sheaves of abelian groups.Proposition 3.21. We make the following assumptions:1. The sites S; S 0 have finite products.2. The morphism of sites f W S ! S 0 preserves finite products.3. We assume that f W ShS 0 ! ShS is exact.Then in D C .Sh Ab S/ there is a natural isomorphismR Hom ShAb S .F; f G/ Š f RHom ShAb S 0.f F;G/:Proof. By [Tam94, Proposition 3.6.7] the conditions on the sites and f imply that thefunctor f is exact. It follows that f preserves injectives (see e.g. [Tam94, Proposition3.6.2]). We choose an injective resolution G ! I . Then f G ! f I is aninjective resolution of f G. It follows thatf RHom ShAb S 0.f F;G/ Š f Hom ShAb S 0.f F;I /Š Hom ShAb S .F; f I / Š RHom ShAb S .F; f G/:Proposition 3.21 is very similar in spirit to Proposition 3.12. On the other hand,we can not deduce 3.12 from 3.21 since the functor f W ShS=U ! S in general doesnot preserve products. In fact, if S has fibre products, then .A ! U/ .B ! U/ D.A U B ! U/. Then f ..A ! U/ .B ! U// Š A U B while f.A !U/ f.B ! U/Š A B.3.3.17 Recall the notations W ; W from 3.3.13.Lemma 3.22. Let W 2 S. For every F 2 ShS and A 2 S we haveHom Sh S .W ; F /.A/ Š W W F .A/:Proof. For A 2 S we consider f W S=A ! S and getHom Sh S .W ; F /.A/ Š Hom Sh S=A .W jA ;F jA /Š Hom Sh S=A .f W ;f F/Š Hom Sh S=A . A .W /; f F/ (by Lemma 3.15)Š Hom Sh S .W ; A f .F //Š A .f F /.W / (16)Š f F.A W ! A/Š F.A W/Š W .f F /.A/Š W . W .F //.A/ (by Lemma 3.15):