@AaronMazel-Gee Sorry, I've asked this question before, and you are not the first to be confused, so somehow the way I'm phrasing this must be wonky. I'm also happy to write more in an email if that is easier, but the short of it is based on the first question.

Fix k a field and A an object of Alg_k. Set F_A : Mod_A -> Set on the category of A-modules, sending B to Der_k(A,B). This functor is representable, for each A-module B, Der_k(A,B) \cong Hom_A(\Omega_{A/k},B).

We may view L_{A/k} as the left Kan extension of Omega_{ - / k} : Alg_k -> A-mod along the inclusion of polynomial algebras. But of course there is some asymmetry to this setup. Hopefully we agree up to here.

The next question is inspired by this discussion. To resolve the asymmetry of the setup, suppose F_A now goes instead Alg_k -> Set, and its represented by an object T_A. Suppose also that T_A is the image of a "nice" functor T : Alg_k -> Alg_k (preserves enough colimits to have a left Kan extension along the inclusion of polynomial algebras). If we assume RF_A : sAlg_k -> sSet is representable, is the representing object the left Kan extension LT_A?

The Toen reference is Lec. 4, Simplicial Commutative Algebras in "Simplicial Methods for Operads", page 161-162, where he claims the (co)representability and cites Goerss-Hopkins and Quillen as references.

Can someone remind me where left/right or (co)Cartesian fibrations up to homotopy are defined? I seem to recall this was done in some work of Lurie but can't find it at the moment. Thanks :)