  We establish a correspondence between consistent comprehension schemes and
complete orthogonal factorisation systems. The comprehensive factorisation of a
functor between small categories arises in this way. Similar factorisation
systems exist for the categories of topological spaces, simplicial sets, small
multicategories and Feynman categories. In each case comprehensive
factorisation induces a natural notion of universal covering, leading to a
Galois-type definition of fundamental group for based objects of the category.
