Let \((I,H,\eta, \varepsilon):{\mathcal C}\to{\mathcal X}\) be an adjunction between categories with pullbacks where \(I:{\mathcal C}\to{\mathcal X}\) is left adjoint to \(H:{\mathcal X}\to{\mathcal C}\). Let \(p:E\to B\) be a given fixed effective descent morphism in \({\mathcal C}\). The adjunction induces an adjunction \((I^E,H^E,\eta^E, \varepsilon^E):{\mathcal C}\downarrow E\to{\mathcal X}\downarrow I(E)\) between slices categories with \(I^E:\mathbb{C}\downarrow E\to \mathbb{X}\downarrow I(E)\) defined by \(I^E(A,\alpha)= (I^E(A), I^E(\alpha))\) and \(I^E(f)= I(f)\). The object \(E\) is said to be \(I\)-admissible if \(\varepsilon^E\) is an isomorphism. Different constructions and examples of admissible objects are given in the paper and, in particular in the so-called ``geometrical categories'' where they are related to connected objects. What is called: ``The fundamental theorem of Galois theory'' in this paper, is expressed as an equivalence of categories, which, under the above admissibility condition, can be written as \({\mathcal S} pl_I(E,p)\simeq{\mathcal X}^{\text{Gal}_I(E,p)}\) where -- \({\mathcal S} pl_I(E,p)\) is the full subcategory of objects \((A,\alpha)\) in \({\mathcal C}\downarrow B\) split over \(p\), -- \({\mathcal X}^{\text{Gal}_I(E, p)}\) is the category of internal actions of the profinite Galois pregroupoid \(\text{Gal}_I(E,p)\) of \((E,p)\). All the notions which are used are defined and illustrated and some examples are given.