X)\big)^{\tilde q/}$.

Similarly, ${\operatorname}{Fun}_K^{{\operatorname}{coCart}}K,X^{q_{K/}})\cong \big({\operatorname}{Fun}_K^{{\operatorname}{coCart}}({K}, X)\big)^{\theta\circ \tilde q/}$. It follows that is an equivalence whenever is.

Combining Proposition \[lem:LimOfInit\] and Lemma \[lem:LimComUnder\] yields the general case:

\[thm:LimInLim\]Let $I$ and $K$ be small simplicial sets, and suppose $p':K\to {{\mathscr{C}\mathrm{at}}}_\infty$ is a functor. Let $\lim p'\in {{\mathscr{C}\mathrm{at}}}_\infty$ denote the limit $\infty$-category and for each $k\in K$, let $\pi_k:\lim p'\to p'(k)$ denote the corresponding functor in the limit cone. Suppose that $q:I\to \lim p'$ is a diagram indexed by $I$, and that

1.  for each vertex $k\in K$, the composite diagram $\pi_k\circ q:I\to p'(k)$ has a (co)limit diagram, and

2.  for each edge $f:k\to k'$ of $K$, the functor $p'(f):p'(k)\to p'(k')$ takes (co)limit diagrams extending $\pi_k\circ q$ to (co)limit diagrams extending $\pi_{k'}\circ q$.

Then:

A

:   there exists a map $\bar q:I^\rhd\to \lim p'$ which extends $q$ and such that each composite $\pi_{k'}\circ \bar q:I^\rhd\to p'(k)$ is a (co)limit (co)cone, and

B

:   an arbitrary extension $\bar q:I^\rhd\to \lim p'$ of $q$ is a (co)limit diagram extending $q$ if and only if each composite $\pi_{k'}\circ \bar q:I^\rhd\to p'(k)$ is a (co)limit diagram extending $\pi_{k'}\circ q$.

In particular, if

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