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Let $C \in Cat$ be an $(\infty,1)$-category.

Let $P$ be the partially ordered subset of full subcategories of $C$.

Is there a (canonical?) functor from the nerve of $P$ to $Cat$? I think the answer must be yes, but I can only define such a functor on $0$ and $1$ simplices of the nerve.

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Fully faithful functors are, in particular, monomorphisms in $\mathrm{Cat}_\infty$. So let $P'$ be the full $\infty$-subcategory of $(\mathrm{Cat}_\infty)_{/C}$ spanned by inclusions of full subcategories. Because these inclusions are monomorphisms, $P'$ is equivalent to the poset $P$, so your desired functor comes from the zig-zag $P\leftarrow P'\rightarrow \mathrm{Cat}_\infty$.

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