Independent of any models or concrete realizations chosen, the notion of $(\infty,1)$-Kan extension is intrinsically determined from just the notions of

In terms of these, for $f : C \to C'$ any (∞,1)-functor and any (∞,1)-category$A$, there is an induced $(\infty,1)$-functor $f^* : Func_{(\infty,1)}(C',A) \to Func_{(\infty,1)}(C,A)$.

The left $(\infty,1)$-Kan extension functor is the left adjoint (∞,1)-functor to $f^*$.

The right $(\infty,1)$-Kan extension functor is the right adjoint (∞,1)-functor to $f^*$.

Given different incarnations of or models for the notion of (∞,1)-category, there are accordingly different incarnations and models of this general abstract prescription.