Tensor product of unit and co-unit in a closed compact category

Consider a compact closed category, i.e., a symmetric monoidal category with a unit $eta$ and co-unit $epsilon$. It seems natural to demand that the tensor product of two units (for different objects) is a unit again (for the tensor product of the objects). That is, we have an equality between the morphisms
$$
1xrightarrow{simeq}
1otimes 1xrightarrow{eta_Aotimes eta_B}
(A^*otimes A)otimes (B^*otimes B) xrightarrow{simeq}
(A^*otimes B^*)otimes (Aotimes B)
$$
and
$$
1xrightarrow{eta_{Aotimes B}} (A^*otimes B^*)otimes (Aotimes B)
$$
$simeq$ stands for some combination of unitors, associators and braidings, I can spell it out if you insist. The analogous equation should hold for the co-unit.

Can this equation be derived from the axioms for compact closed categories? Or is it somehow trivially implied from the way closed compact categories are defined? If not, is there a name for this property, or is it equivalent to some other known property?

Are there closed compact categories for which this equation doesn’t hold? I know that it holds for the compact closed categories of finite vectorspaces, and sets and relations. If I’m trying to define a unit and co-unit for the symmetric monoidal category of super vector spaces, imposing the above equation seems to prevent me from doing that (however, I’m also not sure if super vector spaces can be extended to a compact closed category).