This paper investigates the reflection of universal properties, a cornerstone of category theory, within the formal framework of constructive type theory. Traditional Martin-Löf Type Theory (MLTT) faces challenges in expressing the uniqueness 'up to unique isomorphism' that characterizes such properties due to its intensional nature. We demonstrate that the modern framework of Homotopy Type Theory (HoTT), equipped with the univalence axiom and higher inductive types (HITs), provides a native and conceptually elegant language for this task. By formalizing 'unique existence' as the contractibility of a type of structures, HoTT internalizes the notion of properties holding up to a unique equivalence. We present formalizations of key universal constructions, such as terminal objects, products, and colimits, showing how their universal mapping properties are directly captured by the induction principles of HITs. This approach notifies categorical structures with their type-theoretic counterparts, treating equivalences as paths and thus identifying isomorphic structures at the foundational level. The result is a robust methodology for formalizing abstract mathematics in a constructive and computationally meaningful way, bridging the gap between abstract categorical reasoning and concrete proof-assistant implementations.