Proposition \[lem:LimOfInit\].

Adjunctions and Kan Extensions
------------------------------

### Limits of adjunctions.

Our first application of Theorem \[thm:LimInLim\] is to prove the following result:

\[thm:limadj\] Suppose $$F_k:{\mathcal}{C}_k \rightleftarrows{\mathcal}{D}_k:G_k, \quad k\in K$$ is diagram of adjunctions coherently indexed by a small simplical set $K$, i.e. given by a diagram $(f\dashv g):K\to {\operatorname}{Adj}$ into the $\infty$-category of adjunctions. Let ${\mathcal}{C}=\lim_{k\in K}{\mathcal}{C}_k$ and ${\mathcal}{D}=\lim_{k\in K}{\mathcal}{D}_k,$ and suppose that

-   there is a functor $F:{\mathcal}{C}\to{\mathcal}{D}$ which fits into the cone edge of a diagram $\bar f:K^\lhd\times \Delta^1\to {{\mathscr{C}\mathrm{at}}}_\infty$ extending $f:K\times \Delta^1\to {{\mathscr{C}\mathrm{at}}}_\infty$, and

-   there is a functor $G:{\mathcal}{D}\to {\mathcal}{C}$ which fits into the cone edge of a diagram $\bar g:K^\lhd\times \Delta^1\to {{\mathscr{C}\mathrm{at}}}_\infty$, extending $g:K\times \Delta^1\to {{\mathscr{C}\mathrm{at}}}_\infty$,

such that

1.  the restrictions $\bar f\rvert_{K^\lhd\times\{0\}}$ and $\bar g\rvert_{K^\lhd\times\{1\}}$ are limit cones for ${\mathcal}{C}$, and

2.  the restrictions $\bar f\rvert_{K^\lhd\times\{1\}}$ and $\bar g\rvert_{K^\lhd\times\{0\}}$ are limit cones for ${\mathcal}{D}$.

Then $F$ and $G$ form a pair of adjoint functors $$F:{\mathcal}{C} \rightleftarrows{\mathcal}{D}:G$$

We defer the proof until later: we will first need to give a precise definition of the $\infty$-category, ${\operatorname}{Adj}$, of adjunctions. To do this, we will use the framework for adjunctions of $\infty$-categories in terms of *pairing of $\infty$-categories*, as developed in [@Lurie:ZsNVU6Ru]. For now we give 