Proposition 6.7.12. For an anima \(X \in \An \) and a chain complex \(A \in \D (\Z )\), there is a natural equivalence \[ C^*(X;A) \,\simeq \, \bR \uHom _{\Z }(C_*(X;\Z ), A) \] in \(\D (\Z )\).
Proof. The chain complexes \(C_*(X;\Z )\) are connective. Thus every diagram in the image of the colimit-preserving functor \(C_*(-;\Z )\colon \An \to \D (\Z )_{\geq 0}\) is uniformly bounded below, and Corollary 6.6.36 shows that \(\bR \uHom _{\Z }(C_*(-;\Z ),A)\) preserves limits as a functor on \(\An \catop \). It sends \(\pt \) to \(\bR \uHom _{\Z }(\Z ,A) \simeq A\), so the universal property of \(\An \) identifies it with \(C^*(-;A)\). □
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