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

proof

assume A1: b is terminal ; :: thesis: for a being Object of C ex f being Morphism of a,b st Hom (a,b) = {f}
let a 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 a being Object of C ex f being Morphism of a,b st Hom (a,b) = {f} ; :: thesis: b is terminal
let a be Object of C; :: according to CAT_1:def 18 :: 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