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Duality for topological abelian group stacks and T -duality 2533.3.5 Recall that a functor is called exact if it commutes with limits and colimits.Lemma 3.8. The restriction functor f W ShS ! ShS=Y is exact.Proof. The functor f is a right-adjoint and therefore commutes with limits. Sincecolimits of presheaves are defined object-wise, i.e. for a diagramC ! Pr S;c 7! F cof presheaves we have. p colim c2C F c /.U / D colim c2C F c .U /;it follows from the explicit description of p f , that it commutes with colimits ofpresheaves. Indeed, for a diagram of sheaves F W C ! ShS we havecolim c2C F c D i ]p colim c2C F c :By Lemma 3.7 we get f colim c2C F c Š f i ]p colim c2C F c Š i ] Y p f p colim c2C F c Ši ]p colim c2C p f F c Š colim c2C f F c :3.3.6 Let H 2 ShS and X 2 S.Lemma 3.9. If S has finite products, then for U 2 S we have a natural bijectionHom Sh S .X; H /.U / Š H.U X/:Proof. We have the following chain of isomorphismsHom Sh S .X; H /.U / Š Hom Sh S=U .X jU ;H jU /Š Hom Sh S=U .U X ! U ;H jU / (by Lemma 3.6)Š H.U X/:We will need the explicit description of this bijection. We define a map‰ W Hom Sh S .X; H /.U / ! H.U X/as follows. An element h 2 Hom Sh S=U .X; H /.U / D Hom Sh S .X jU ;H jU / induces amap Q hW X jU .U X ! U/ ! H jU .U X ! U/. Now we have X jU .U X !U/ D X.U X/ D Map.U X; X/ and H jU .U X ! U/ D H.U X/. Letpr X W U X ! X be the projection. We define ‰.h/ WD Q h.pr X /.We now define ˆW H.UX/ ! Hom Sh S .X; H /.U /. Let g 2 H.UX/. For each.e W V ! U/2 S=U we must define a map Og .V !U/ W X jU .V ! U/! H jU .V ! U/.Note that X jU .V ! U/D X.V / D Map.V; X/ ı . Let x 2 Map.V; X/. Then we havean induced map .e; x/W V ! U X. We define ‰.g/ WD H.e; x/.g/. One checksthat this construction is functorial in e and therefore defines a morphism of sheaves.A straight forward calculation shows that ‰ and ˆ are inverses to each other andinduce the bijection above.