@DylanWilson I cannot find it in Goerss-Jardine, but can't you send a simplex Δ^n→X to the space of sections Δ^n→Y? Then functoriality is given by precomposition. This should work, at least when Y→X is a Kan fibration (no idea what happens if Y→X is a random map)
Actually, that would be a functor (Δ/X)^{op}→wF, if you want Δ/X→wF you need to send Δ^n→X to Y×_X Δ^n. I prefer the first because it gives us explicitely fibrant models, but the second one works too.
Huh, they have changed that formulation somewhat in the edition I have. There they say that the map from the hocolim is a weak equivalence and does not mention the thing about the diagonal. Oh well.
The functor is defined by just pulling back, i.e the functor that Denis gives in his most recent message.
In the second edition (the one you can access on SpringerLink, or well, buy it) , the lemma is on pg. 235 and there it says that the map holim f^{-1} -> E is a weak equivalence
I do not know too much about things like these, so I can make silly mistakes
suppose i have two varieties X and Y. how close is $Fun^{ex}(Coh(X),Coh(Y))$ to the set/space of maps in one direction or the other? i always get confused, because there are just so many darn adjoints around. but i feel like general exact functors shouldn't be related to any sorts of maps: the usual "integral transforms" story refers to left-adjoint functors.
