The Defect Functor of a Homomorphism and Direct Unions

We will study commuting properties of the defect functor \(\text {Def}_{\beta }=\text {Coker}\text {Hom}_{\mathcal {C}}(\beta ,-)\) associate to a homomorphism ß in a finitely presented category. As an application, we characterize objects M such that \(\text {Ext}^{1}_{\mathcal {C}}(M,-)\) commutes with direct unions (i.e. direct limits of monomorphisms), assuming that \(\mathcal {C}\) has a generator which is a direct sum of finitely presented projective objects.