Kerodon

"Corollary 01F3 guarantees that $\theta$ is also an isofibration", it is a bit confused how to apply Corollary 01F3 here. However, to prove Proposition 05HN, it is enough for $\theta$ to be an equivalence of $\infty$-categories.

...say by Proposition 0285 (2), $\theta$ will be an equivalence of inner fibrations over $\mathrm{Fun}(\mathrm{N}_\bullet(\pmb{\Delta}_+^{\mathrm{op}}),\mathcal{C})$. We consider the map induced by $\theta$ on the fibre at $\check{\mathrm{C}}(X/Y)\in\mathrm{Fun}(\mathrm{N}_\bullet(\pmb{\Delta}_+^{\mathrm{op}}),\mathcal{C})$, which will be an equivalence of $\infty$-categories.