I have made more explicit statement and proof that functorial fibrant replacement in a left Bousfield localized model category is a model for the derived adjunction unit, here

]]>I have added statement and proof (here) that given a left Bousfield localization, then local fibrations between local objects are global fibrations.

(I am assuming functorial factorization, which is of course provided under the generic assumption of the entry that we are localizing a cofibrantly generated model category. But probably one can get the statement more generally.)

]]>added the statement that left Bousfield localization is functorial with respect to Quillen equivalences, *here*.

Based on some feedback that I got behind the scenes, I have now rearranged the material, to make it more light-weight at the beginning.

the section Properties now has the central properties one can deduce for a Bousfield localization that do not require combinatorial model category technology

the section Existence of ... has the existence result for the combinatorial case pretty much as before, but minus the material that has been moved to "Properties"

and the Definition section is now such that it downplays the "derived hom" and instead tends to arrange things such that they are properly cofibrant or fibrant . Because it was suggested to me that invoking the derived hom concept makes this look, to the non-expert reader- much more arcane than it is.

I just hope that the interconnecting logic between the paragraphs is now still consistent. I fddled a bit with adjusting the statements to the slightly changed assimptions.

]]>I completed the existece proof with the bit showing the accessibility of the S-local weak equivalences.

Then I added a section prerequisite for the proof that lists briefly all the material at other entries, previously just having been linked to, that enters the proof.

]]>finished typing part 1) and 2) of the proof of the existence theorem at Bousfield localization of model categories

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