$\infty$-category $\lim_{k\in K}{\mathcal}{C}_k$ also has all (co)limits of shape $I$. After providing an alternate proof of this result (cf. Theorem \[thm:LimInLim\]), we extend their result to show that if each ${\mathcal}{C}_k$ possesses all Kan extensions along a functor ${\mathcal}{I}\to{\mathcal}{I}'$, and each functor ${\mathcal}{C}_k\to {\mathcal}{C}_{k'}$ preserves those Kan extensions, then the limit $\infty$-category $\lim_{k\in K}{\mathcal}{C}_k$ also has all Kan extensions along ${\mathcal}{I}\to{\mathcal}{I}'$ (cf. Corollary \[cor:KanClosed\]). More generally, suppose that $$F_k:{\mathcal}{C}_k \rightleftarrows{\mathcal}{D}_k:G_k, \quad k\in K$$ is diagram of adjunctions coherently indexed by a small simplical set $K$, then we prove there is an adjunction $$\lim_{k\in K}F_k:\lim_{k\in K}{\mathcal}{C}_k \rightleftarrows\lim_{k\in K}{\mathcal}{D}_k:\lim_{k\in K}G_k$$ between the corresponding limit $\infty$-categories (cf. Theorem \[thm:limadj\]).

Acknowledgements
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We would like to thank Owen Gwilliam, Theo Johnson-Freyd, Claudia Scheimbauer, and Alan Weinstein for many important discussions surrounding the content of this paper. We would also like to thank Thomas Nikolaus for a very helpful introduction to descent theory, and Omar Antol[í]{}n Camarena for a number of helpful conversations about higher category theory. The author was supported by an NSF Postdoctoral Fellowship DMS-1204779.

Notation
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We generally use the notation and terminology developed by Lurie (cf. [@Lurie:2009un]). In particular, by an *$\infty$-category*, we mean a quasicategory, i.e. a simplicial set satisfying 