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.
Best Answer
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.