let C be Category; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: 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 ;

:: thesis: verum

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; :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: 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 ;

:: thesis: verum