Canonical lcn_gFun n := [gFun by lcn_cont n].
Canonical lcn_mgFun n := [mgFun by fun _ G H => @lcnS _ n G H].
Section UpperCentralFunctor.
Variable n : nat.
Implicit Type gT : finGroupType.

Lemma ucn_bigcprod n I r P (F : I -> {set gT}) G :
\big[cprod/1]_(i <- r | P i) F i = G ->
\big[cprod/1]_(i <- r | P i) 'Z_n(F i) = 'Z_n(G).