If the abstract map is a bijection, the map from each U onto V carries an n-simplex onto an n-simplex. Dimension is preserved, giving a homeomorphism. Therefore the two topological complexes are equivalent.
If an abstract complex has two topological instantiations, the identity map on vertices (a bijection) proves the spaces are homeomorphic. There is but one topological space, up to isomorphism, for each abstract complex. The embedding of K into Ej is a default representation.
A simplicial map that carries a copmlex into itself is sometimes called a pasting map, as it pastes vertices together.
If you like category theory, the geometric realization of an abstract complex is a functor from sets of finite sets (closed under subset) into topological spaces, where functions on vertices become continuous maps. If we restrict topological spaces to simplicial complexes, and if functions are simplicial maps, the categories are equivalent. In other words, the functor admits an inverse functor, which focuses on the vertices of a topological complex.