Chain duality V (Ex)

Let $\Aa$$<wikitex>; Let $\Aa$ be an additive category with chain duality $(T,e)$. Show that under the isomorphism $$ T_{M,TM} \colon M \otimes_{\Aa} TM \rightarrow TM \otimes_{\Aa} M $$ we have $$ T_{M,TM} (\textup{id}) = e_{M}. $$ Observe that $e$ can be defined this way if we have already defined $T$ and $T_{M,N}$. </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises with solution]]\Aa$ be an additive category with chain duality $(T,e)$$(T,e)$. Show that under the isomorphism

$\displaystyle T_{M,TM} \colon M \otimes_{\Aa} TM \rightarrow TM \otimes_{\Aa} M$ $$\displaystyle T_{M,TM} \colon M \otimes_{\Aa} TM \rightarrow TM \otimes_{\Aa} M$$

we have

$\displaystyle T_{M,TM} (\textup{id}) = e_{M}.$ $$\displaystyle T_{M,TM} (\textup{id}) = e_{M}.$$

Observe that $e$$e$ can be defined this way if we have already defined $T$$T$ and $T_{M,N}$$T_{M,N}$.