of a coCartesian edge over $k\to k'$ starting at $x$.

Given a second coCartesian fibration $p_0:X_0\to K$ corresponding to a diagram $p_0':K\to {{\mathscr{C}\mathrm{at}}}_\infty$, Gepner, Haugseng, and Nikolaus identify the $\infty$-category of natural transformations $${\operatorname}{Nat}_{K}(p_0',p')\cong {\operatorname}{Fun}_K^{{\operatorname}{coCart}}(X_0,X)$$ with the $\infty$-category of coCartesian maps $X_0\to X$, i.e. those maps $X_0\to X$ over $K$ which preserve the coCartesian edges (cf. [@Gepner:2015ww Proposition 6.9]). In particular, Lurie shows that an elegant model of $\lim p'$ is the $\infty$-category $$\lim p'\cong {\operatorname}{Nat}_{K}(\Delta_\ast,p')\cong {\operatorname}{Fun}_K^{{\operatorname}{coCart}}(K,X)$$ of coCartesian sections of $p$ [@Lurie:2009un Corollary 3.3.3.2] (note that every edge in the trivial fibration $K\to  K$ is coCartesian).

Let ${{\mathscr{C}\mathrm{at}}}_\infty^{{\operatorname}{Str}}$ denote the ordinary category whose objects are (small) $\infty$-categories, and suppose that $C$ is a (small) category. Given a strict functor $p':C\to {{\mathscr{C}\mathrm{at}}}_\infty^{{\operatorname}{Str}}$, the corresponding coCartesian fibration can be computed via the relative nerve construction (cf. [@Lurie:2009un § 3.2.5]). Forming the resulting $\infty$-category of coCartesian sections, one sees that a model for $\lim p'$ is the simplicial set whose $k$ simplices consist of the following data:

-   for every functor $(x,y):[n]\to C\times [k]$, a choice of $n$-simplex, $\tau_{(x,y)}:\Delta^n\to p'\big(x(n)\big)$, such that

1.  for every $f:[m]\to [n]$, the following diagram commutes: $$\begin{tikzpicture}
        {\matrix (m) [matrix of math nodes, row sep=3em,
      column sep=3em, 