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Note on the universality and the functoriality of the perfect F-locality

In "Frobenius Categories versus Brauer Blocks" we have proved some universality of the so-called localizing functor associated with a Frobenius $P$-category $F$, where $P$ is a finite $p$-group, with respect to the coherent $F$-localities $(τ,L,π)$ such that the contravariant functor $Ker (π)$ maps any subgroup of $P$ to an Abelian $p$-group. The purpose of this Note is both to move from the localizing functor to the perfect locality associated with $F$ and to remove the Abelian hypothesis in the target. As a consequence, we get the functoriality for the perfect localities in the strongest form, improving the Theorem 9.15 in "Existence, uniqueness and functoriality of the perfect locality over a Frobenius $P$-category".