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282 U. Bunke, T. Schick, M. Spitzweck, and A. ThomBy Lemma 3.8 the restriction functor is exact and therefore commutes with colimits.We getHom ShAb S .colim P .F /; I /.A/ Š Hom ShAb S=A..colim P .F // jA ;I jA /Š Hom ShAb S=A.colim.P .F / jA /; I jA /Š lim I Hom ShAb S=A.P .F / jA ;I jA /Š lim . I Hom ShAb S .P .F /; I /.A//Š .lim I Hom ShAb S .P .F /; I //.A/;where in the last step we use that a limit of sheaves is defined object-wise. In otherwordsHom ShAb S .colim F;I / Š Hom ShAb S .colim P .F /; I / Š lim I Hom ShAb S .P .F /; I /Applying Lemma 4.19 we see that I Hom ShAb S .P .F /; I / 2 Hom Cat .I; Sh Ab S/ isinjective, andRHom ShAb S .colim F;H/ Š lim I Hom ShAb S .P .F /; I /Š R lim I Hom ShAb S .P .F /; I /Š R lim R I Hom ShAb S .F; H /:4.3.4 Lemma 4.21 implies the existence of s spectral sequence with second pageE p;q2Š R p lim R qI Hom ShAb S .F; H / (26)which converges to GrR Hom ShAb S .colim F;H/.Lemma 4.22. Let I be a category and U 2 S. Then we have a commutative diagramD C ..Sh Ab S/ I op R lim/ D C .Sh Ab S/R.U;::: /D C .Ab I op /R limR.U;::: / D C .Ab/.Proof. Since the limit of a diagram of sheaves is defined object-wise we have .U;:::/ılim D lim ı .U;:::/. We show that the two compositions R.U;:::/ı R lim ,R lim ıR.U;:::/are both isomorphic to R..U;:::/ılim/ by showing that lim and .U;:::/preserve injectives.The left-adjoint of the functor limW .Sh Ab S/ I op! Sh Ab S is the constant diagramfunctor C ::: W Sh Ab S ! .Sh Ab S/ I op which is exact. Therefore lim preserves injectives.The left-adjoint of the functor .U;:::/is the functor ˝Z Z.U /. Since Z.U /is a sheaf of flat Z-modules (it is torsion-free) this functor is exact (object-wise andtherefore on diagrams). It follows that .U;:::/preserves injectives.