The authors unify, in generalizing, the results of \textit{H. Herrlich} [Math. Z. 150, 101-110 (1976; Zbl 0319.18001)] and \textit{G. C. L. Brümmer} and \textit{R.-E. Hoffmann} [Lect. Notes Math. 540, 136-151 (1976; Zbl 0334.54001)] on injective fibre-small concrete categories over a base category and injective hulls with three of their variants. For the full panoply, including characterization of the objects which have an injective hull, one specifies a base category \(\underset \tilde{} X\), and a nicely limit-closed (``limit coherent'') family \({\mathcal S}\) of finite sources in \(\underset \tilde{} X\), to describe injectives in the concrete categories over \(\underset \tilde{} X\) that have initial lifts of structured \({\mathcal S}\)-sources. The mere characterization of injectives is proved without the finiteness restriction. Examples of \textit{E. Nelson} in \(\sigma\)-semilattices [Can. Math. Bull. 18, 387-392 (1975; Zbl 0323.06006)] show that without that restriction, injective hulls present a much more delicate problem.