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@Urs: I answered your query over at homotopy category of an (infinity,1)-category. The Ho(SSet) enrichment is given by the lax monoidal functor SSet -> Ho(SSet).
Been a while since I looked at that entry. I have edited now a few more thing, such as mentioning the relation to the homotopy category of a category with weak equivalences.
That reminds me that our entry simplicial localization is badly in need of more attention.
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