Chain duality III (Ex)

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Let F : \Aa \rightarrow \Aa' be a functor of additive categories with chain duality. Show that the assignment

\displaystyle   M \otimes_{\Aa} N \rightarrow F(M) \otimes_{\Aa'} F(N)

given by

\displaystyle   \varphi \colon TM \rightarrow N \quad \mapsto \quad F (\varphi) \circ G(M) \colon T' F (M) \rightarrow FT(M) \rightarrow F(N)

induces a \Zz_2-equivariant chain map

\displaystyle   C \otimes_{\Aa} C \rightarrow F(C) \otimes_{\Aa'} F(C)

for any C \in \Bb(\Aa).

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