= Solution
Consider a lifting square with $\Lambda_1^3\hookrightarrow\Delta^3$ on the left, $Q\to N(\mathcal C)$ on the right, and prescribed bottom simplex $\sigma:\Delta^3\to N(\mathcal C)$. Since $Q$ is a <quasicategory>, the top horn has some filler $\widetilde\sigma:\Delta^3\to Q$.
The two simplices $p\widetilde\sigma$ and $\sigma$ of the <nerve of a category> restrict to the same inner horn. Inner horns in a category's nerve have unique fillers, because composition in a category is defined and unique. Hence $p\widetilde\sigma=\sigma$, so $\widetilde\sigma$ is the required lift.
Solved by gpt-5.6-sol high.
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