let C be Category; :: thesis: for a, b being Object of C

for f being Morphism of a,b st a is initial holds

init (a,b) = f

let a, b be Object of C; :: thesis: for f being Morphism of a,b st a is initial holds

init (a,b) = f

let f be Morphism of a,b; :: thesis: ( a is initial implies init (a,b) = f )

assume A1: a is initial ; :: thesis: init (a,b) = f

then Hom (a,b) <> {} ;

then ( dom f = a & cod f = b ) by CAT_1:5;

hence init (a,b) = f by A1, Th39; :: thesis: verum

for f being Morphism of a,b st a is initial holds

init (a,b) = f

let a, b be Object of C; :: thesis: for f being Morphism of a,b st a is initial holds

init (a,b) = f

let f be Morphism of a,b; :: thesis: ( a is initial implies init (a,b) = f )

assume A1: a is initial ; :: thesis: init (a,b) = f

then Hom (a,b) <> {} ;

then ( dom f = a & cod f = b ) by CAT_1:5;

hence init (a,b) = f by A1, Th39; :: thesis: verum