T0-reflection and injective hulls of fibre spaces

We give a characterization of injective (with respect to the class of embeddings) topological fibre spaces using their T0-reflection, that turns out to be injective itself. We then prove that the existence of an injective hull of (X,f) in the category Top/B of topological fibre spaces is equivalent to the existence of an injective hull of its T0-reflection (X0,f0) in Top/B0 (and in the category Top0/B0 of T0 topological fibre spaces).