Dear All Notation as in [Hartshorne]: Let f:X->Y be a continuous map between topological spaces. The adjunction Hom_X(f^{-1}(.), .) <-> Hom_Y(., f_*(.)) for sheaves is classical: For a sheaf F on Y and a sheaf G on X, the adjunction gives a natural isomorphism Hom_X(f^{-1}F, G) <-> Hom_Y(F, f_*G) of vector spaces (or abelian groups). I concocted an internal adjunction (hopefully correct) f_*hom_X(f^{-1}(.), .) <-> hom_Y(., f_*(.)) of sheaves on Y: for a sheaf F on Y and a sheaf G on X, a natural isomorphism f_*hom_X(f^{-1}F, G) <-> hom_Y(F, f_*G) of sheaves on Y. Is there a place in the literature which discusses the situation, in particular, spells out that internal adjunction (if it exists)? For ringed spaces and F locally free, the isomorphism f_*hom_{O_X}(f^*F, G) <-> hom_{O_Y}(F, f_*G) is straightforward. Some Mathoverflow pages deal with that internal adjunction but I did not find a reference there. Does the issue make sense for morphisms between sites? Is there such an internal adjunction for a morphism between sites? Many thanks in advance Best Johannes ---------- You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. Leave group: https://outlook.office365.com/groups/groupsubscription?source=EscalatedMessage&action=leave&smtp=categories%40mq.edu.au&bO=true&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27