an immediate corollary:

\[cor:KanClosed\] Suppose that $\delta:I\to I'$ is a morphism of simplicial sets. Let ${{\mathscr{C}\mathrm{at}}}_\infty^\delta\subset{{\mathscr{C}\mathrm{at}}}_\infty$ be the subcategory consisting of

-   those $\infty$-categories ${\mathcal}{C}$ which admit left (right) Kan extensions along $\delta$ for any functor $f:I\to {\mathcal}{C}$, and

-   those functors ${\mathcal}{C}\to{\mathcal}{C}'$ which preserve left (right) Kan extensions along $\delta$.

Then ${{\mathscr{C}\mathrm{at}}}_\infty^\delta\subset{{\mathscr{C}\mathrm{at}}}_\infty$ is closed under (small) limits.

Let $\phi:{{\mathscr{C}\mathrm{at}}}_\infty\to{\operatorname}{Fun}(\Delta^1,{{\mathscr{C}\mathrm{at}}}_\infty)$ be the functor which sends an $\infty$-category ${\mathcal}{C}$ to the pullback-functor $$\label{eq:rKanExt}{\operatorname}{Fun}(I,{\mathcal}{C})\leftarrow {\operatorname}{Fun}(I',{\mathcal}{C}):\delta^*.$$ Evaluating $\phi$ at either endpoint of $\Delta^1$ $$\begin{aligned}
{\operatorname}{ev}_{\{0\}}\circ \phi:{{\mathscr{C}\mathrm{at}}}_\infty\xrightarrow{{\mathcal}{C}\mapsto {\operatorname}{Fun}(I',{\mathcal}{C})}{{\mathscr{C}\mathrm{at}}}_\infty\\
{\operatorname}{ev}_{\{1\}}\circ \phi:{{\mathscr{C}\mathrm{at}}}_\infty\xrightarrow{{\mathcal}{C}\mapsto {\operatorname}{Fun}(I,{\mathcal}{C})}{{\mathscr{C}\mathrm{at}}}_\infty\end{aligned}$$ results in continuous functors (they are right adjoints). Therefore [@Lurie:2009un Corollary 5.1.2.3] implies that $\phi$ is continuous.

Now ${\mathcal}{C}$ admits left Kan extensions along $\delta$ if and only if is a right adjoint. In particular, we may identify ${{\mathscr{C}\mathrm{at}}}_\infty^\delta$ with the pullback $$\begin{tikzpicture}
    {\matrix (m) [matrix of math nodes, row sep=3em,
  column sep=3em, text height=1.5ex, text depth=0.25ex]}{{{\mathscr{C}\mathrm{at}}}_\infty^\delta& {{\mathscr{C}\mathrm{at}}}_\infty\\{\operatorname}{Adj}&{\operatorname}{Fun}(\Delta^1,{{\mathscr{C}\mathrm{at}}}_\infty)\\};
    \draw[->] (m-1-2) -- node {$\phi$} (m-2-2);
    \draw[->] (m-2-1) -- node {$R$} (m-2-2);
    \draw[dashed,->] (m-1-1) -- (m-1-2);
    \draw[dashed,->] (m-1-1) -- (m-2-1);
\end{tikzpicture}$$ Where $R:{\operatorname}{Adj}\to{\operatorname}{Fun}(\Delta^1,{{\mathscr{C}\mathrm{at}}}_\infty)$ is the functor which sends an adjunction $(F\dashv G)$ to its right adjoint $G$. By Theorem \[thm:limadj\], the functor $R$ is continuous. Therefore, by Theorem \[thm:LimInLim\], ${{\mathscr{C}\mathrm{at}}}_\infty^\delta$ admits all small limits and 