For simple graphs G and H, the Hom complex \(\textrm{Hom}(G,H)\) is a polyhedral complex whose vertices are the graph homomorphisms \(G\rightarrow H\) and whose edges connect the pairs of homomorphisms which differ in a single vertex of G. Hom complexes play an important role in an algebro-topological approach to the graph coloring problem. It is known that \(\textrm{Hom}(G,H)\) is homotopy equivalent to a disjoint union of points and circles when both G and H are cycles. We generalize this known result by showing that the same holds whenever G is connected and H is a cycle. To this end, we explicitly construct the universal cover of each connected component of \(\textrm{Hom}(G,H)\) and prove that it is contractible. Additionally, we provide a simple criterion to determine whether the connected component containing a given homomorphism is homotopy equivalent to a point or circle.
