Let $\mathcal S_L$ be any accessible reflective subcategory of the $\infty$-category of spaces. In his book, Farjoun discusses "fiberwise $L$-localization" of a map of spaces $E \to B$, i.e. a factorization $E \to \bar E \to B$ such that the fiber of $\bar E \to B$ is in $\mathcal S_L$ and $E \to \bar E$ is $\mathcal S_L$-local. Chataur and Scherer show (Thm 4.3) that a fiberwise localization always exists.

Farjoun claims (paragraph 1.F.5) that any fiberwise localization enjoys a universal property, but he's not quite precise on what that means -- he says (with slight paraphrase) "the map $E \to \bar E$ as a map over $B$ is universal among all maps of $E$ to spaces over $B$ that have homotopy fiber in $\mathcal S_L$".

I'm not quite clear on what "universal" means here, but surely it should at least imply that $\bar E$ is unique up to equivalence over $B$. But I'm having trouble seeing that. So I ask,

**Question:** Let $\mathcal S_L$ be any accessible reflective subcategory of the $\infty$-category of spaces. Let $E \to B$ be a map and let $\bar E \to B$, $\bar E' \to B$ be two fiberwise $L$-localizations of $E \to B$. Then are $\bar E \to B$, $\bar E' \to B$ equivalent over $B$ (and under $E$)? More strongly, is the space of fiberwise localizations of $E \to B$ contractible?