  In this paper we give a unified approach in categorical setting to the
problem of finding the Galois closure of a finite cover, which includes as
special cases the familiar finite separable field extensions, finite unramified
covers of a connected undirected graph, finite covering spaces of a locally
connected topological space, finite \'etale covers of a smooth projective
irreducible algebraic variety, and finite covers of normal varieties. We
present two algorithms whose outputs are shown to be desired Galois closures.
An upper bound of the degree of the Galois closure under each algorithm is also
obtained.
