Here is an explicit example of a full, additive subcategory of an abelian category that is itself abelian but the inclusion functor is not exact. Let "k" be a field, formula_34 the algebra of upper-triangular formula_35 matrices over "k", and formula_36 the category of finite-dimensional formula_34-modules. Then each formula_36 is an abelian category and we have an inclusion functor formula_39 identifying the simple projective, simple injective and indecomposable projective-injective modules. The essential image of "I" is a full, additive subcategory, but "I" is not exact.