let C be Category; :: thesis: for a being Object of C holds
( a is initial iff for b being Object of C ex f being Morphism of a,b st Hom (a,b) = {f} )
let a be Object of C; :: thesis: ( a is initial iff for b being Object of C ex f being Morphism of a,b st Hom (a,b) = {f} )
thus ( a is initial implies for b being Object of C ex f being Morphism of a,b st Hom (a,b) = {f} ) :: thesis: ( ( for b being Object of C ex f being Morphism of a,b st Hom (a,b) = {f} ) implies a is initial )

proof

assume A1: a is initial ; :: thesis: for b being Object of C ex f being Morphism of a,b st Hom (a,b) = {f}
let b be Object of C; :: thesis: ex f being Morphism of a,b st Hom (a,b) = {f}
consider f being Morphism of a,b such that
A2: for g being Morphism of a,b holds f = g by A1;
take f ; :: thesis: Hom (a,b) = {f}
thus Hom (a,b) = {f} by A2, Th7, A1; :: thesis: verum

end;

assume A3: for b being Object of C ex f being Morphism of a,b st Hom (a,b) = {f} ; :: thesis: a is initial
let b be Object of C; :: according to CAT_1:def 19 :: thesis: ( Hom (a,b) <> {} & ex f being Morphism of a,b st
for g being Morphism of a,b holds f = g )
consider f being Morphism of a,b such that
A4: Hom (a,b) = {f} by A3;
thus Hom (a,b) <> {} by A4; :: thesis: ex f being Morphism of a,b st
for g being Morphism of a,b holds f = g
take f ; :: thesis: for g being Morphism of a,b holds f = g
thus for g being Morphism of a,b holds f = g by A4, Th6; :: thesis: verum