Corollary 6.6.36. Assume in addition that \(\Aa \) satisfies \((AB3)\), \((AB4)\), \((AB3^*)\), and \((AB4^*)\). For every \(C \in \D ^-(\Aa )\), the functor \(\bR \uHom (C,-)\colon \D (\Aa ) \to \D (\Aa )\) preserves all small limits. For every \(E \in \D (\Aa )\), the functor \(\bR \uHom (-,E)\) sends colimits of uniformly bounded below diagrams in \(\D ^-(\Aa )\) to limits.
Proof. The functor \(\bR \uHom (C,-) \simeq \uHom (P,-)\), for \(P\) a projective resolution of \(C\), is exact and satisfies \(\uHom (P, \prod _i E_i) \cong \prod _i \uHom (P, E_i)\) degreewise. It therefore preserves all limits.
In the first variable, the functor is exact and sends coproducts of uniformly bounded below families to products: if \(P^i\) is a projective resolution of \(C^i\), chosen with a common lower bound, then \(\bigoplus _i P^i\) is a projective resolution of \(\bigoplus _i C^i\), and \[ \uHom \Bigl (\bigoplus _i P^i, E\Bigr ) \; \cong \; \prod _i \uHom (P^i,E) \] degreewise. Together with exactness, this gives the assertion for uniformly bounded below colimits. □
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