The cohomological comparison arising from the associated abelian object

We make explicit some conditions on a semi-abelian category D such that, for any abelian group A in D and any object Y in D, the cohomology group homomorphisms with coefficients in A, induced by the inclusion of the abelian objects of D at the level of the slice category D/Y, are actually isomorphisms. These conditions hold in particular when D is the category Gp of groups, and this allows us to give a new insight on the Eilenberg-Mac Lane cohomology of groups. They hold also when D is the category K-Lie of Lie-algebras.