the empty set, in which case we have the following variant of [@Lurie:2009un Proposition 2.4.4.9]:

\[lem:LimOfInit\] Given a functor $p':K\to {{\mathscr{C}\mathrm{at}}}_\infty$, if

1.  for each $k\in K$, the $\infty$-category $p'(k)$ admits an initial object $t\in X_k$, and

2.  for each edge $k\to k'$ in $K$ the functor $p'(k)\to p'(k')$ preserves initial objects,

then

A

:   the limit $\infty$-category $\lim p'$ admits an initial object $t_\infty$, and

B

:   an object $t\in \lim p'$ is initial if and only if for each $k\in K$, the object $\pi_k(t)\in X_k$ is initial where $\pi_k:\lim p'\to p'(k)$ is the functor appearing in the limit cone.

We will find it easier to model our functor $p':K\to {{\mathscr{C}\mathrm{at}}}_\infty$ in terms of a *Cartesian fibration* $p:X\to K^{op}$ (rather than a *coCartesian fibration*).

Let $p:X\to K^{op}$ be a Cartesian fibration of simplicial sets classified by the functor $p':K\to {{\mathscr{C}\mathrm{at}}}_\infty$. By assumption

1.  for each $k\in K$, the $\infty$-category $X_k\cong p'(k)$ admits an initial object $t\in X_k$, and

2.  for each $p$-Cartesian edge $f:t'\to t$ over $p(f):k'\to k$ the object $t'\in X_{k'}$ is initial whenever $t\in X_k$ is,

Let $X'\subseteq X$ be the simplical subset spanned by those vertices $t\in X$ which are initial objects of $X_{p(t)}\cong p'\circ p(t)$. 