Then (as we shall show), every edge $f:t'\to t$ in $X'$ is $p$-Cartesian (when seen as an edge of $X$). To see this suppose that $f:t'\to t$ is such an edge. Let $f':t''\to t$ be a $p$-Cartesian edge in $X$ over $p(f)$; then (cf. [@Lurie:2009un Remark 2.4.1.4.]) there exists a 2-simplex $\sigma:\Delta^2\to X$ such that $$\sigma(\Delta^{\{1,2\}})=f',\quad \sigma(\Delta^{\{0,2\}})=f,\quad\text{and}\quad p\big(\sigma(\Delta^{\{0,1\}})\big)=p(s_0(t')),$$ where $s_0:K_0\to K_1$ is the degeneracy map.

By assumption $t'',t'\in X_{p(s)}$ are both initial, and hence $\sigma(\Delta^{\{0,1\}})\in X_{p(t')}$ is an equivalence. In particular $\sigma(\Delta^{\{0,1\}})\in X$ is a $p$-Cartesian morphism. It follows from [@Lurie:2009un Proposition 2.4.1.7.] that $f=\sigma(\Delta^{\{0,2\}})$ is $p$-Cartesian.

Now, by [@Lurie:2009un Proposition 2.4.4.9.], there exists a section $t_\infty:K^{op}\to X'$; and by the previous discussion, $t_\infty$ is a Cartesian section.

Now $\lim p'\cong {\operatorname}{Fun}_{K^{op}}^{{\operatorname}{Cart}}({K^{op}},X)$ is the $\infty$-category of Cartesian sections of $p$ (cf. [@Lurie:2009un Corollary 3.3.3.2]). Thus, we can identify $t_\infty$ with an element of $\lim p'$. We now claim that $t_\infty\in \lim p'$ is an initial object: Notice that $\lim p'$ is the full subcategory of ${\operatorname}{Fun}_{K^{op}}(K^{op},X)$ spanned by the Cartesian sections. Suppose we have a diagram $$\begin{tikzpicture}
        {\matrix (m) [matrix of math nodes, row sep=3em,
  column sep=3em, text height=1.5ex, text depth=0.25ex]}{\partial \Delta^n & {\operatorname}{Fun}_{K^{op}}^{{\operatorname}{Cart}}({K^{op}},X) & {\operatorname}{Fun}_{K^{op}}({K^{op}},X)\\ \Delta^n & &\\};
        \draw[->] 