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

proof

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

end;

assume A3: for a being Object of C ex f being Morphism of b,a st Hom (b,a) = {f} ; :: thesis: b is initial
let a be Object of C; :: according to CAT_8:def 6 :: thesis: ( Hom (b,a) <> {} & ex f being Morphism of b,a st
for g being Morphism of b,a holds f = g )
consider f being Morphism of b,a such that
A4: Hom (b,a) = {f} by A3;
thus Hom (b,a) <> {} by A4; :: thesis: ex f being Morphism of b,a st
for g being Morphism of b,a holds f = g
take f ; :: thesis: for g being Morphism of b,a holds f = g
thus for g being Morphism of b,a holds f = g by A4, Th6; :: thesis: verum