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 $I$.