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It seems to me that it should be mostly correct. A Kan extension of itself along will factor through the quotient map , since is an ordinary category, and therefore it will be just (the nerve of) the ordinary derived functor of F composed with the quotient map , and thus not containing any "higher information". But probably what you really want is to take the Kan extension of the composite along ; that ought to remember the higher homotopies. But I don't know whether this is written down anywhere.
do I understand you correctly, that you want C(infinity) to denote the simplicial localization of the category with weak equivalences C?
I suppose you have seen the query box discussion at derived functor?
Allow me to assume that we have a bit more information than just weak equivalences available, namely a nice model category structure. Then the (oo,1)-category presented by C is the simplicial localization restricted to the cofibrant-fibrant objects (as described by the Dwyer-Kan theorem recalled by the big diagram at (infinity,1)-categorical hom-space).
Typically the functors that are of interest respect this simplicial enrichement. As Mike desribes in the query box at derived functor, in that case the "correct" extension of F to the (oo,1)-category presented by C is its restriction as a simplicially enriched functor to the fibrant-cofibrant objects (or equivalently its precomposition with a fibrant-cofibrant replacement functor).
C doesn't have to be a model category in order to go to the infinity-case; any category with weak equivalences has a simplicial localization. What you need a model category for (or, more generally, a deformation retract of a homotopical category) is to define derived functors in the "useful" way as being given by precomposition with a "replacement" functor, rather than in the "un-useful" way as a Kan extension.
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Kan extensions are "un-useful" in what way?
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<p>As a way of characterizing derived functors. See the query box discussion at <a href="https://ncatlab.org/nlab/show/derived+functor">derived functor</a>.</p>
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If you know of any theorems about derived functors, defined as Kan extensions without making use of the fact that in good cases (such as model categories) they can be computed with fibrant/cofibrant replacement, or even of any theorem about a particular derived functor which uses the fact that it is a Kan extension, I'd be interested to hear it. The only thing I can think of which comes close is the fact that a given functor has at most one left derived functor, independent of whatever model structure you choose to compute it with, but that can also just as easily be proven directly.
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