  Let $p$ be a prime, $P$ a finite p-group and $\cal F$ a Frobenius
$P$-category. In "Existence, uniqueness and functoriality of the perfect
locality over a Frobenius $P$-category", Algebra Colloquium, 23(2016) 541-622,
we also claimed the uniqueness of the partial perfect locality $\cal L^{\frak
X}$ over any up-closed set $\frak X$ of $\cal F$-selfcentralizing subgroups of
$P$, but recently Bob Oliver exhibit some counter-examples, demanding some
revision of our arguments. In this Note we show that, up to replacing the
perfect localities by the "extendable" perfect localities over any up-closed
set $\frak X$ of $\cal F$-selfcentralizing subgroups of $P$, our arguments are
correct, still proving the existence and the uniqueness of the perfect $\cal
F^{\rm sc}$-locality, since it is "extendable". We take advantage to simplify
some of our arguments.
