let C1, C2 be Category; :: thesis: for F being Functor of C1,C2
for a, b being Object of C1 holds (Upsilon F) . (homsym a,b) = homsym (F . a),(F . b)
let F be Functor of C1,C2; :: thesis: for a, b being Object of C1 holds (Upsilon F) . (homsym a,b) = homsym (F . a),(F . b)
let a, b be Object of C1; :: thesis: (Upsilon F) . (homsym a,b) = homsym (F . a),(F . b)
A1: dom (Obj F) = the carrier of C1 by FUNCT_2:def 1;
thus (Upsilon F) . (homsym a,b) = [0 ,((Obj F) * ((homsym a,b) `2 ))] by Def13
.= [0 ,((Obj F) * <*a,b*>)] by MCART_1:7
.= [0 ,<*((Obj F) . a),((Obj F) . b)*>] by A1, FINSEQ_2:145
.= [0 ,<*(F . a),((Obj F) . b)*>] by CAT_1:def 20
.= homsym (F . a),(F . b) by CAT_1:def 20 ; :: thesis: verum