text height=1.5ex, text depth=0.25ex]}{\Delta^m & &p'\big( x(f(m))\big)\\ \Delta^n&&p'\big( x(n)\big)\\};
        \draw[->] (m-1-1) -- node {$f$} (m-2-1);
        \draw[->] (m-1-1) -- node {$\tau_{f^*(x,y)}$} (m-1-3);
        \draw[->] (m-2-1) -- node {$\tau_{(x,y)}$} (m-2-3);
        \draw[->] (m-1-3) -- node {$p'\big[x\big(f(m)\to n\big)\big]$} (m-2-3);
     \end{tikzpicture}$$ and

2.  whenever $y(i)=y(j)$, then $\tau_{(x,y)}(\Delta^{\{i,j\}})\in p'\big(x(n)\big)$ is an equivalence.

(co)Limits in Limit $\infty$-categories
---------------------------------------

Let $I$ be a second (small) simplicial set, and suppose that

1.  for each vertex $k\in K$, the $\infty$-category $p'(k)$ admits (co)limits for all diagrams indexed by $I$.

2.  for each edge $(k\to k')\in K$, the functor $p'(k)\to p(k')$ preserves (co)limits for all diagrams indexed by $I$.

then Riehl and Verity [@Riehl:2014ut] have shown that the limit $\infty$-category $\lim p' \in {{\mathscr{C}\mathrm{at}}}_\infty$ admits (co)limits for all diagrams indexed by $I$, and that those (co)limits are preserved by the functors in the limit cone. We now provide an alternate proof of this result, based on Lurie’s (co)Cartesian fibrations.

To disambiguate our presentation, we will prove our results only for *colimits* (rather than *limits*) in the limit $\infty$-category $\lim p'$; the duality between colimits in $\lim p'$ and limits in $(\lim p')^{op}$ imply that the corresponding results hold equally for limits as well. We begin with the special case where $I$ is 