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When is a Ch-enriched (enriched over the category of chain complexes) functor $F: A \to B$ determined by the bimodule $A^{op} \otimes B \to Ch_{dg}$, $(a,b) \mapsto Hom_B(F(a),b)$. Or in general how much of $F$ do we remember? This might be standard stuff but would like a reference.

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What do you mean by a Ch-enriched functor? –  Fernando Muro Jun 2 '12 at 0:19
Presumably, a functor enriched in the category of chain complexes. –  Tom Leinster Jun 2 '12 at 0:47
If I understand the question correctly, then the answer is "always" by the enriched Yoneda lemma, section 2.4 in Kelly's "Basic concepts of enriched category theory". –  Mike Shulman Jun 2 '12 at 5:33

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