universal property unique up to unique isomorphism

A universal property defines an object - if it exists - up to unique isomorphism

Consider a category and some universal property, that both , and satisfy. Then by assumption there exist unique morphism and , usually such that following diagram commutes

Furthermore, itself satisfies the universal property and makes the following diagram commute

Thus by composition of morphism and uniqueness of the mediating morphism , it follows, that

Hence are isomorphisms and furthermore the isomorphism between is by construction uniquely defined