{{\mathscr{C}\mathrm{at}}}_\infty$ is a diagram with cone point ${\mathcal}{C}\in {{\mathscr{C}\mathrm{at}}}_\infty$. For any vertex $k\in K^\lhd$, let $\pi_k:{\mathcal}{C}\to p'(k)$ denote the corresponding functor in the cone. Given any diagram $q:I\to {\mathcal}{C}$, since the formation of undercategories is natural, there exists a diagram $(\bar p')^{q/}:K^\lhd\to {{\mathscr{C}\mathrm{at}}}_\infty$ indexing the undercategories: $$(\bar p')^{q/}(k)\cong \big(\bar p'(k)\big)^{\pi_k\circ q/},$$ for all $k\in K^{\lhd}$, along with a natural transformation $(\bar p')^{q/}\to \bar p'$ which restricts at every $k\in K^\lhd$ to the canonical functors $\big(\bar p'(k)\big)^{\pi_k\circ q/}\to \bar p'(k)$.

\[lem:LimComUnder\] If $\bar p':K^{\lhd}\to {{\mathscr{C}\mathrm{at}}}_\infty$ is a limit diagram, then so is $(\bar p')^{q/}:K^\lhd\to {{\mathscr{C}\mathrm{at}}}_\infty$.

Let $\bar p:\bar X \to K^\lhd$ be a coCartesian fibration classified by $\bar p'$; and let $\ast\in (K^{op})^\lhd$ denote the cone point. Since $\{\ast\}^\sharp\subseteq \big(K^\lhd\big)^\sharp$ is marked anodyne, we have a natural equivalence of the $\infty$-category of coCartesian sections: $${\operatorname}{Fun}_{K^\lhd}^{{\operatorname}{coCart}}(K^\lhd,\bar X)\xrightarrow{\cong}{\operatorname}{Fun}_{K^\lhd}^{{\operatorname}{coCart}}(\{\ast\},\bar X)\cong \bar X\times_{K^\lhd}\{\ast\}\cong {\mathcal}{C}.$$ Therefore, we may lift $q:I\to {\mathcal}{C}$ to a diagram $$\tilde q:I\to {\operatorname}{Fun}_{K^\lhd}^{{\operatorname}{coCart}}(K^\lhd,\bar X)$$ in a homotopically unique way. Choose a factorization $I \to I' \to  {\operatorname}{Fun}_{K^\lhd}^{{\operatorname}{coCart}}(K^\lhd,\bar X)$ of $\tilde q$, where $I \to I'$ is inner anodyne (and therefore a categorical equivalence) and $I' \to  {\operatorname}{Fun}_{K^\lhd}^{{\operatorname}{coCart}}(K^\lhd,\bar X)$ is an inner fibration (so that $I'$ is an $\infty$-category). The map $I \to 