Lemma 6.6.30 (Tensor–hom adjunction for chain complexes). For chain complexes \(A,B,C\in \Ch (\Aa )\), there is a natural isomorphism of chain complexes \[ \uHom (A\otimes B,C) \cong \uHom (A,\uHom (B,C)), \] which induces the tensor-hom adjunction \[ \Hom _{\Ch (\Aa )}(A \otimes B, C) \,\cong \, \Hom _{\Ch (\Aa )}(A, \uHom (B, C)). \]
Proof. A proof is given in the standalone supplementary material. □
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