I'$ is a categorical equivalence, and therefore cofinal. We are free to replace $I$ by $I'$, and may thereby assume that $I$ is an $\infty$-category.

Given two morphisms of simplicial sets $Y\to K^\lhd$, and $Z\to K^\lhd$, recall that $Z\diamond_{K^\lhd} Y$ denotes the relative (alternate) join of the simplicial sets $Z$ and $Y$, $$Z\diamond_{K^\lhd} Y:= Z\coprod_{Z\times_{K^\lhd} Y\times \{0\}}(Z\times_{K^\lhd} Y\times \Delta^1)\coprod_{Z\times_{K^\lhd} Y\times \{1\}}Y$$ (cf. [@Lurie:2009un § 4.2.2] and [@Joyal:2008wr]).

Let $q_{K^\lhd}:I\times K^\lhd\to \bar X$ denote the composite $$I\times K^\lhd\xrightarrow{q\times {\operatorname}{Id}_{K^\lhd}}{\operatorname}{Fun}_{K^\lhd}^{{\operatorname}{coCart}}(K^\lhd,\bar X)\times K^\lhd\xrightarrow{{\operatorname}{ev}} \bar X.$$ As in [@Lurie:2009un § 4.2.2] we define $\bar X^{q_{K^\lhd}/}\to K^\lhd$ to be the simplicial set satisfying the universal property that for any morphism of simplicial sets $Y\to K^\lhd$, commutative diagrams of the form $$\begin{tikzpicture}
    {\matrix (m) [matrix of math nodes, row sep=3em,
  column sep=3em, text height=1.5ex, text depth=0.25ex]}{I\times K^\lhd &\bar X\\(I\times K^\lhd)\diamond_{K^\lhd} Y& K^\lhd\\};
    \draw[->] (m-1-1) -- node {$q_{K^\lhd}$} (m-1-2);
    \draw[right hook->] (m-1-1) --  (m-2-1);
    \draw[->] (m-1-2) -- node {$\bar p$} (m-2-2);
    \draw[->] (m-2-1) -- (m-2-2);
    \draw[->] (m-2-1) --  (m-1-2);
\end{tikzpicture},$$ correspond to diagrams of the form $$\begin{tikzpicture}\node (y) at (-2,1) {$Y$};
\node (q) at (2,1) {$\bar X^{q_{K^\lhd}/}$};
\node (css) at (0,0) {$K^\lhd$};
\draw[->] (y) -- (q);
\draw[->] (y) -- (css);
\draw[->] (q) -- (css);
\end{tikzpicture}$$

Note that 