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?