Homotopy-coherent replacement of diagrams in quasi-categories In Emily Riehl's book Categorical Homotopy Theory, a proposition (16.3.1) attributed to Cordier-Porter states that if $\underline{\mathcal{C}}$ is a fibrant simplicial category, and if $F : \mathcal{A} \to \underline{\mathcal{C}}$ is a so-called homotopy coherent diagram, then given a family of equivalences $Fa \to Ga$ in $\underline{\mathcal{C}}$, then there is some way to extend all of this data to a new homotopy coherent diagram $G : \mathcal{A} \to \underline{\mathcal{C}}$, as well as a (homotopy-coherent) natural transformation $F \Rightarrow G$ extending the family of maps given. There is a similar statement (16.3.2) for modyfiyinh natural transformations.
I am looking for similar statements in the setting of $(\infty,1)$-categories à la Joyal-Lurie. I guess the kind of statement I am looking for would be that given a simplicial set $K$, an $\infty$-category $\mathcal{C}$, a functor $F : K \to \mathcal{C}$ and given a family of equivalences $Fa \to Ga$ for all $0$-simplex $a$ of $K$, then there is a functor $G : K \to \mathcal{C}$ as well as a natural transformation $F \Rightarrow G$ extending this data.
I have never seen in the litterature such a kind of statement in this setting so that I fear that it might be "too good to be true", and maybe there needs to be some additional assumptions to be made on $K$ or on $F$, I'm fine with this.
Given the equivalence between quasi-categories and simplicial categories, I guess the statement holds if $F$ arises as the homotopy-coherent nerve of some homotopy-coherent diagram. This pdf seems to suggest (page 10) that this would hold for instance if $K$ is the nerve of a $1$-category. It would be good to see it stated precisely somewhere.
Any reference for that kind of statement (maybe with additionnal assumptions on K) would be appreciated.
 A: It is true.
For every simplicial set $K$, the exponential object $[K, \mathcal{C}]$ is a quasicategory if $\mathcal{C}$ is.
Let $K_0$ be the set of vertices of $K$, considered as a discrete simplicial set.
The inclusion $K_0 \hookrightarrow K$ is a monomorphism, so the induced functor $[K, \mathcal{C}] \to [K_0, \mathcal{C}]$ is an isofibration of quasicategories.
But an equivalence in $[K_0, \mathcal{C}]$ is precisely a $K_0$-indexed family of equivalences, so this says that for any functor $F : K \to \mathcal{C}$ and any $K_0$-indexed family of equivalences $F a \to G a$ ($a \in K_0$), there is a functor $G : K \to \mathcal{C}$ whose objects are the specified objects and a natural equivalence $F \Rightarrow G$ whose components are the specified equivalences.
The hard work is in the statement that $[-, \mathcal{C}]$ sends monomorphisms to isofibrations.
This is basically the fact that the Joyal model structure is cartesian, plus the explicit identification of fibrations between fibrant objects as isofibrations of quasicategories.
