The $f$-localization I mean is the one described and studied in detail in the book by E. D. Farjoun; $L_f$ is a homotopy idempotent functor which associates to each space $X$ an $f$-equivalence $X\to L_f(X)$ where $L_f(X)$ is $f$-local.

$f$-localization has a kind of uniqueness: if $F$ is some other coaugmented functor with the property that $F(X)$ is $f$-local for every $X$ (I'm happy to assume that $F = L_g$ for some map $g$), then there is a commutative square of functors and natural transformations, which I don't know how to draw here. The square would show that the composites $$ id \xrightarrow{\iota} L_f \xrightarrow{L_f(j)} L_f\circ F \qquad \mathrm{and} \qquad id \xrightarrow{j} F \xrightarrow{\iota_F} L_f \circ F $$ are equal. And $\iota_F$ is a weak equivalence for every space $X$; thus $F$ factors through $L_f$ `up to weak equivalence'.

My Question: Suppose $X\to Y$ is an $f$-equivalence; then $L_f(X) \to L_f(Y)$ is a weak equivalence; does it follow that $(L_f\circ F)(X) \to (L_f\circ F)(Y)$ is a weak equivalence?