On the Homeomorphism of Continuous Mappings

Denote by $\mathcal{C}(X)$ the partially ordered (PO) set of all continuous epimorphisms of a space $X$ under the natural identification of homeomorphic epimorphisms. The following homeomorphism theorem for bicompacta is implicitly contained in Magill's 1968 paper: two bicompacta $X$ and $Y$ are homeomorphic if and only if the PO sets $\mathcal{C}(X)$ and $\mathcal{C}(Y)$ are isomorphic.
In the present paper, Magill's theorem is extended to the category of mappings in which the role of bicompacta is played by perfect mappings. The results are obtained in two versions, namely, in the category $\mathit{TOP}_Z$ (of triangular commutative diagrams) and in the category $\mathit{MAP}$ (of quadrangular commutative diagrams).