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Let $C$ be an abelian category, with a projective separator $k$. Assume that $C$ has a duality, that's a functor $\ast:C\to C^{\text{op}}$ together with a natural isomorphism $\tau:1\to \ast\ast$ such that $\tau_V^\ast=\tau_{V^\ast}^{-1}$ for each object $V$.

Let $q:V\to V^{*}$ be a morphism; $q$ can be interpreted as a bilinear form on $V$. To say that $q=q^{\ast}\circ \tau_V$ means that $q$ is symmetric.

Question: is true that each $q:k\to k^\ast$ is symmetric?

Remark: The answer to the question is yes in $R$-(left) modules category with $R$ a field (pheraps also if $R$ is commutative unitary ring) with $V^\ast=\text{Hom}(V,R)$. If $R$ is not commutative, then $V^\ast=\text{Hom}(V,R)$ is not a $R$-(left) module, hence $V\mapsto V^\ast$ doesn't define a duality in the sense of previous definition.

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