Question is the following:

Is the functor $H^n_{dR}:\text{Man}\rightarrow \text{Set}$ a sheaf with respect to open cover topology on $\text{Man}$?

More generally, are cohomology functors sheaves in general (in any reasonably non trivial Grothendieck topology)?

I am also interested in cohomology functors that arise in Algebriac geometry/topology.

Is there a way of sheafification in this setup?

I have nothing much to support this question, this is completely out of curiosity.

Edit : I am also interested in answers/references related to the comment of Piotr Achinger; that reads

"in what way is cohomology a sheaf" leads one to notions like $\infty$-topoi etc.

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