The structure of Bethe vectors for generalised models associated with the rational and trigonometric R-matrix is investigated. The Bethe vectors in terms of two-component and multi-component models are described. Consequently, their structure in terms of local variables and operators is provided. This, as a consequence, proves the equivalence of coordinate and algebraic Bethe ansatzes for the Heisenberg spin chains. Hermitian conjugation of the elements of the monodromy matrix for the spin chains is studied.