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This article presents several results concerning the exactness of covariant and contravariant Hom functors and their derived functors in a balanced abelian category \( A \). In particular: (i) The functors \( Hom_A (X,-) \) and \( Hom_A (-,X) \) are left exact, and become exact if and only if \( X \) is projective (resp. injective). (ii) The functors \( Hom_Comp(A)(X,-) \) and \( Hom_Comp(A)(-,X) \) on the category of complexes \( Comp(A) \) preserve this behavior. (iii) The homological functors \( H_n(X,-) \) and \( H_n(-,X) \) are constructed for all \( n Z \). (iv) For projective \( X \), the connecting morphism \( λ_n: H_n(X,-)((T,γ)) → H_n+1(X,-)((Y,α)) \) allows \( H_n(X,-) \) to send short exact sequences in \( Comp(A) \) into long exact sequences in \( Ab \). (v) Similarly, for injective \( X \), the morphism \( δ_n: H_n(-,X)((Y,α)) → H_n+1(-,X)((T,γ)) \) shows that \( H_n(-,X) \) also preserves long exact sequences. , number=4, journal=European Journal of Pure and Applied Mathematics, year=2025, month=Nov., pages=6640