In A characterization of simplicial localization functors and a discussion of DK equivalences Barwick and Kan state that, while there is no preferred localization functor from relative categories to simplicial categories, there is a preferred relativization functor from simplicial categories to relative categories. It is not apparent to me in which sense the relativization functor is "preferred". No alternative to this construction ever seems to be discussed, nor is there any discussion about how this construction is in any way canonical or "forced" upon us. Thus I would like to ask:

In which sense is the relativization functor "preferred"?

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    $\begingroup$ Looking at it, I don't think you should ascribe any special meaning to "preferred" here. I think they are just saying they know many examples of localization functors, but there is only one kind of relativization functor that they ever work with in practice. $\endgroup$ Dec 10 '16 at 14:46

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