by $I$, and

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

then the limit $\infty$-category $\lim p'$ admits all (co)limits of shape $I$, and the functors $\pi_k:\lim p'\to p'(k)$ fitting into the limit cone preserve all (co)limits of shape $I$.

Let $\bar p':K^\lhd\to {{\mathscr{C}\mathrm{at}}}_\infty$ be a limit cone extending $p'$ which maps the cone point $\infty\in K^\lhd$ to $\lim p'$ and the cone edge $\infty\to k$ to $\pi_k$ for each $k\in K^\lhd$. Let $(\bar p')^{q/}:K^\lhd\to {{\mathscr{C}\mathrm{at}}}_\infty$ denote corresponding diagram of undercategories, as in Lemma \[lem:LimComUnder\]. Then $(\bar p')^{q/}:K^\lhd\to {{\mathscr{C}\mathrm{at}}}_\infty$ is a limit cone which (by assumptions (1) and (2)) satisfies the assumptions for Proposition \[lem:LimOfInit\]. Now for any $k\in K^\lhd$, the $\infty$-category of diagrams $I^\rhd\to \bar p'(k)$ extending $\pi_k\circ q$ is equivalent to the undercategory $\big(\bar p'(k)\big)^{\pi_k\circ q}$, and this equivalence identifies colimit diagrams with initial objects of the undercategory. Moreover, for any edge $k\to k'$ in $K^\lhd$, the functor $\big(\bar p'(k)\big)^{\pi_k\circ q/}\to \big(\bar p'(k')\big)^{\pi_{k'}\circ q/}$ preserves initial objects if and only if the functor $\bar p'(k)\to \bar p'(k')$ takes colimits diagrams extending $\pi_k\circ q$ to colimit diagrams extending $\pi_{k'}\circ q$.

Therefore statements **A** and **B** follow from 