Lifting covering maps

Let X̃p→X be a covering projection. We use p♯ :C(Y,X̃)→C(Y,X) to denote the map defined by p♯(f̃)=p∘f̃, where spaces of continuous maps are endowed with the compact-open topology. We prove that if Y is Hausdorff and contractible, or if Y is either a compact CW-complex or a graph with finitely many components and p♯ is onto, then p♯ is a covering projection.