let C be Category; :: thesis: for I being Indexing of C
for T being TargetCat of I holds pr2 (I -functor C,T) = I `2
let I be Indexing of C; :: thesis: for T being TargetCat of I holds pr2 (I -functor C,T) = I `2
let T be TargetCat of I; :: thesis: pr2 (I -functor C,T) = I `2
A1: dom (I -functor C,T) = the carrier' of C by FUNCT_2:def 1;

A2: now

let x be set ; :: thesis: ( x in the carrier' of C implies (pr2 (I -functor C,T)) . x = (I `2 ) . x )
assume x in the carrier' of C ; :: thesis: (pr2 (I -functor C,T)) . x = (I `2 ) . x
then reconsider f = x as Morphism of C ;
(I -functor C,T) . f = [[((I `1 ) . (dom f)),((I `1 ) . (cod f))],((I `2 ) . f)] by Def11;
then ((I -functor C,T) . x) `2 = (I `2 ) . f by MCART_1:7;
hence (pr2 (I -functor C,T)) . x = (I `2 ) . x by A1, MCART_1:def 13; :: thesis: verum

end;

( dom (pr2 (I -functor C,T)) = dom (I -functor C,T) & dom (I `2 ) = the carrier' of C ) by MCART_1:def 13, PARTFUN1:def 4;
hence pr2 (I -functor C,T) = I `2 by A1, A2, FUNCT_1:9; :: thesis: verum