2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/106363We show that certain canonical realizations of the complexes Hom(G,H) and Hom_+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these complexes is itself again a complex, and exhibit several well-known objects that arise as cells or subcomplexes of such projected Hom-complexes: the dissections of a convex polygon into k-gons, Postnikov's generalized permutohedra, staircase triangulations, the complex dual to the lower faces of a cyclic polytope, and the graph of weak compositions of an integer into a fixed number of summands.23 pages, 5 figures; improved exposition; accepted for publication in JCTACombinatoricsMetric Geometry52B11 (Primary) 05C99, 52B70 (Secondary)text