let C be Category; for a, b being Object of C
for f being Morphism of C st a is initial & dom f = a & cod f = b holds
init (a,b) = f
let a, b be Object of C; for f being Morphism of C st a is initial & dom f = a & cod f = b holds
init (a,b) = f
let f be Morphism of C; ( a is initial & dom f = a & cod f = b implies init (a,b) = f )
assume that
A1:
a is initial
and
A2:
( dom f = a & cod f = b )
; init (a,b) = f
consider h being Morphism of a,b such that
A3:
for g being Morphism of a,b holds h = g
by A1;
f is Morphism of a,b
by A2, CAT_1:4;
hence f =
h
by A3
.=
init (a,b)
by A3
;
verum