Let $k$ be a field. Let $C$ be a small category and assume that for every $i\geq 0$ we have a functor $H^i:C\rightarrow FinDimVect_k$. Assume that there is a function $dim:Obj(C)\rightarrow \mathbb{Z}_{\geq 0}$ such that for every $c\in Obj(C)$ and every $0\leq i \leq 2 dim(c)$ we have a functorial perfect pairing $H^i(c)\times H^{2dim(c)-i}(c)\rightarrow H^{2dim(c)}(c)$.

Now assume we have two objects $c_1$ and $c_2$ such that $dim(c_1)=dim(c_2)=n$ and such that for some $i\geq 0$ the spaces $H^i(c_1)$ and $H^i(c_2)$ are non-isomorphic. Is it true that there can not simultaneously exist morphisms $f:c_1\rightarrow c_2$ and $g:c_2\rightarrow c_1$ inducing isomorphisms in $H^{2n}$?