It is well known that, for two functors $F,G : I \to C$ for $I,C$ some $\infty$-categories, the property that a map $\phi: F \to G$ is an equivalence can be checked locally on $I$. Namely, if $\phi(i) : F(i) \to G(i)$ is an equivalence for every $i \in I$, then $\phi$ is an equivalence as a morphism in the 
$\infty$-category $Fun(I,C)$. 


Is the same true for adjointability? Namely: 


Let $I$ be an $\infty$-category, and let $F,G : I \to Cat\_\infty$ be two functors into the $(\infty,1)$ category of $\infty$-categories. Let $\phi : F \to G$.
Suppose that, for each $i \in I$, $\phi(i) : F(i) \to G(i)$ admits left adjoint 
$L\_{\phi(i)}$. Then, there nessecarily exists a map $\psi: G \to F$ restricting to $L\_{\phi(i)}$ on every object? Is it essentially unique? does it satisfy some relative version of the property satisfied by left adjoints? 


My motivation for believing in it is that the space of choices of the Left adjoint is contractible, so there should be no obstructions to glue them over $I$. 



$(\infty,2)$-Categorical Analogue of the Local Nature of Equivalences
**TL DR:** That is not enough. If you let $\psi\_i:G(i)\to F(i)$ be the left adjoint of $\phi\_i$ you also need the condition that for every $f:i\to j$ the canonical morphism
$$\psi\_jG(f)\to F(f)\psi\_i$$
adjoint to $G(f)\to G(f)\phi\_i\psi\_i\cong \phi\_jF(f)\psi\_i$, is an equivalence.


The key word to remember here is *relative adjunctions*. If we unstraighten your functors $F$ and $G$, we end up with two cartesian fibrations $E\to I^{op}$ and $E'\to I^{op}$ and a map of cartesian fibrations $\phi:E\to E'$ such that for each $i\in I$ the functor $\phi\_i:E\_i\to E'\_i$ is a right adjoint. Then, by theorem 7.3.2.6 in Higher Algebra, there is a relative left adjoint $\psi$. The result you are after would follow if and only if $\psi$ were a map of cartesian fibrations (i.e. iff it sends cartesian edges to cartesian edges). Unwrapping the various definitions, this is exactly the condition I wrote above.


To see a counterexample when the condition is not satisfied, pick two categories $C\_0,C\_1$ with an initial object and a functor $f:C\_0\to C\_1$ that does *not* preserve the initial object (e.g. let $C\_0$ be finite pointed sets, $C\_1$ be finite sets and $f$ the functor that forgets the pointing). This assembles to a functor $F:\Delta^1\to \mathrm{Cat}\_\infty$.


Now let $G:\Delta^1\to \mathrm{Cat}\_\infty$ be the constant functor at $\*$. There is an obvious natural transformation $\phi:F\to G$ and $\phi\_0$ and $\phi\_1$ have both left adjoints $\psi\_0$ and $\psi\_1$ (the inclusion of the initial object). However these do not assemble into a natural transformation $\psi:G\to F$ because $\psi\_1$ is not naturally equivalent to $f\psi\_0$.


