The defect functor of a homomorphism and direct unions

We will study commuting properties of the defect functor $\mathrm{Dev}_β=\mathrm{Coker}\mathrm{Hom}_\mathcal{C}(β,-)$ associate to a homomorphism $β$ in a finitely presented category. As an application, we characterize objects $M$ such that $\mathrm{Ext}^1_\mathcal{C}(M,-)$ 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.

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