In Higher Topos Theory, Lurie describes the simplicial nerve functor taking as input a simplicially enriched category $\mathcal C$ and outputting the ∞-category $\mathrm{N}(\mathcal C)$.
The construction is functorial in the following way: if $F \colon \mathcal C \to \mathcal D$ is a simplicially enriched functor between two simplicially enriched categories, then $\mathrm{N}(F)$ is a functor from $\mathrm{N}(\mathcal C)$ to $\mathrm{N}(\mathcal D)$. This defines a functor from the 1-category of simplicially enriched categories and simplically enriched functors to the category of simplicial sets.
What happens to the simplicial natural transformations?
I believe that on several occasions the following is used: simplicial natural isomorphisms are sent to invertible natural transformations. Is there a quick way to see this or a reference for it?