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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I have now typed out full proof of the adjoint equivalence.
It follows (not in the entry yet) with K. Brown’s lemma that for simplicial presheaves this adjoint equivalence is a Quillen adjunction for the projective model structure and the slice of that.
Still need to think about full proof that this is a Quillen equivalence. But need to interrupt now.
Added mentioning (here) of the application to the proof of cohesion of global- over G-equivariant homotopy theory.
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