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280 U. Bunke, T. Schick, M. Spitzweck, and A. Thomwhich fits into the adjoint pairZ I˝F W Hom Cat .I op ; Sh Ab S/ ,W Sh Ab S W I Hom ShAb S .F;:::/; (25)where for G 2 Hom Cat .I op ; Sh Ab S/ the symbol R I G ˝ F 2 Sh Ab S denotes the coendof the functor I op I ! Sh Ab , .i; j / 7! G.i/˝ F.j/; correspondingly R Idenotes theend of the appropriate functor. Indeed we have for all A 2 Hom Cat .I op ; Sh Ab S/ andB 2 Sh Ab S a natural isomorphismI ZIHom ShAb S.ZA ˝ F;B/ ŠopI Hom ShAb S .A ˝ F;B/IZIŠop Hom ShAb S .A; I Hom ShAb S .F; B//IŠ Hom HomCat .I op ;Sh Ab S/.A; I Hom ShAb S .F; B//The functor I Hom ShAb S .F;:::/ is therefore also left-exact and admits a rightderivedversion.4.3.3 We say that P 2 Hom Cat .I; Sh Ab S/ is I -free if there exists a collection offlat sheaves .U.l// l2I , U.l/ 2 Sh Ab S, such that P.j/ D L l!jU.l/, and P.i !j/W L l!i U.l/ ! L l!jU.l/ maps the summand U.l/ at l ! i identically to thesummand U.l/ at l ! i ! j .Lemma 4.19. If P 2 Hom Cat .I; Sh Ab S/ is I -free, and if J 2 Sh Ab S is injective, thenI Hom ShAb S .P; J / 2 Hom Cat.I op ; Sh Ab S/ is injective.Proof. We consider an exact sequence.A W 0 ! A 0 ! A 1 ! A 2 ! 0/in Hom Cat .I op ; Sh Ab S/. Then by Equation (25) we haveHom HomCat .I op ;Sh Ab S/.A ; I Hom ShAb S .P; J // Š Hom Sh Ab S Z IA ˝ P;JExactness in Hom Cat .I op ; Sh Ab S/ is defined object-wise [Tam94, Theorem 0.1.3.1].Therefore 0 ! A 0 .i/ ! A 1 .i/ ! A 2 .i/ ! 0 is an exact complex of sheaves for alli 2 I . Since P.j/is flat for all j 2 I the complex A .i/ ˝ P.j/is exact for all pairs.i; j / 2 I I . The complex of sheaves R I A ˝ P is the complex of push-outs alongthe exact sequence of diagramsF.i!j/2Mor.I /.j / ˝ P.i/id˝P.i!j/ FA j 2I.j / ˝ P.j/A:A .i!j/˝idF i2I.i/ ˝ P.i/.A