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I am puzzled by this page, where it discusses morphisms between ringed spaces. In particular, it requires a morphism $f^\sharp : O_Y\to f_*O_X$ to be a homomorphism of rings and then comments that the adjointness does not imply that the transpose $f_\sharp : f^*O_Y \to O_X$ is a morphism of rings. But both Hakim and Lurie define morphisms of ringed spaces to be given by morphisms of rings $f_\sharp : f^*O_Y\to O_X$. Furthermore, the Stacks project actually claims that morphisms of rings $f^*O_Y \to O_X$ are the same, by adjointness, as morphisms of rings $O_Y \to f_*O_X$. Who is wrong here?
I have deleted this passage from the entry:
However, in the case of ringed sites, a morphism $G\to f_* F$ is required to be morphism of sheaves of rings, whereas the adjunction does not guarantee that the corresponding morphism $f^* G\to F$ of sheaves of sets over $X$ is a morphism of sheaves of rings over $X$.
Because, this contradicts standard sources (as pointed out above) and the author of that line (revision 1 in 2009) does not remember why he wrote it (here).
Of course, better than just deleting the sentence would be to replace it by an actual analysis of what’s going on.
added pointer to:
The following remark 13.1.1 (p. 316) explicitly says that the two adjoint forms of comorphisms are equivalent.
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