reconsider OM = id [:the carrier of A,the carrier of A:] as bifunction of the carrier of A,the carrier of A ;
set MM = id the Arrows of A;
id the Arrows of A is MSUnTrans of OM,the Arrows of A,the Arrows of A

proof

per cases ( [:the carrier of A,the carrier of A:] <> {} or [:the carrier of A,the carrier of A:] = {} ) ;
:: according to FUNCTOR0:def 4

case [:the carrier of A,the carrier of A:] <> {} ; :: thesis: ex I2' being non empty set ex B' being ManySortedSet of ex f' being Function of [:the carrier of A,the carrier of A:],I2' st
( OM = f' & the Arrows of A = B' & id the Arrows of A is ManySortedFunction of the Arrows of A,B' * f' )

then reconsider I2' = [:the carrier of A,the carrier of A:] as non empty set ;
reconsider A' = the Arrows of A as ManySortedSet of ;
reconsider f' = OM as Function of [:the carrier of A,the carrier of A:],I2' ;
take I2' ; :: thesis: ex B' being ManySortedSet of ex f' being Function of [:the carrier of A,the carrier of A:],I2' st
( OM = f' & the Arrows of A = B' & id the Arrows of A is ManySortedFunction of the Arrows of A,B' * f' )
take A' ; :: thesis: ex f' being Function of [:the carrier of A,the carrier of A:],I2' st
( OM = f' & the Arrows of A = A' & id the Arrows of A is ManySortedFunction of the Arrows of A,A' * f' )
take f' ; :: thesis: ( OM = f' & the Arrows of A = A' & id the Arrows of A is ManySortedFunction of the Arrows of A,A' * f' )
thus ( OM = f' & the Arrows of A = A' ) ; :: thesis: id the Arrows of A is ManySortedFunction of the Arrows of A,A' * f'
thus id the Arrows of A is ManySortedFunction of the Arrows of A,A' * f' :: thesis: verum

proof

let i be set ; :: according to PBOOLE:def 18 :: thesis: ( not i in [:the carrier of A,the carrier of A:] or (id the Arrows of A) . i is Element of bool [:(the Arrows of A . i),((A' * f') . i):] )
assume A1: i in [:the carrier of A,the carrier of A:] ; :: thesis: (id the Arrows of A) . i is Element of bool [:(the Arrows of A . i),((A' * f') . i):]
then (A' * f') . i = A' . (f' . i) by FUNCT_2:21
.= the Arrows of A . i by A1, FUNCT_1:35 ;
hence (id the Arrows of A) . i is Element of bool [:(the Arrows of A . i),((A' * f') . i):] by A1, PBOOLE:def 18; :: thesis: verum

end;

end;

case A2: [:the carrier of A,the carrier of A:] = {} ; :: thesis: id the Arrows of A = [0] [:the carrier of A,the carrier of A:]

then id the Arrows of A = {} by PBOOLE:134;
hence id the Arrows of A = [0] [:the carrier of A,the carrier of A:] by A2; :: thesis: verum

end;

end;

end;

then reconsider MM = id the Arrows of A as MSUnTrans of OM,the Arrows of A,the Arrows of A ;
take FunctorStr(# OM,MM #) ; :: thesis: ( the ObjectMap of FunctorStr(# OM,MM #) = id [:the carrier of A,the carrier of A:] & the MorphMap of FunctorStr(# OM,MM #) = id the Arrows of A )
thus ( the ObjectMap of FunctorStr(# OM,MM #) = id [:the carrier of A,the carrier of A:] & the MorphMap of FunctorStr(# OM,MM #) = id the Arrows of A ) ; :: thesis: verum