  We construct universal monoidal categories of topological tensor supermodules
over the Lie superalgebras $\mathfrak{gl}(V\oplus \Pi V)$ and
$\mathfrak{osp}(V\oplus \Pi V)$ associated with a Tate space $V$. Here $V\oplus
\Pi V$ is a $\mathbb{Z}/2\mathbb{Z}$-graded topological vector space whose even
and odd parts are isomorphic to $V$. We discuss the purely even case first, by
introducing monoidal categories$\widehat{\mathbf{T}}_{\mathfrak{gl}(V)}$,
$\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and
$\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$, and show that these categories are
anti-equivalent to respective previously studied categories
$\mathbb{T}_{\mathfrak{gl}(V)}$, $\mathbb{T}_{\mathfrak{o}(V)}$,
$\mathbb{T}_{\mathfrak{sp}(V)}$. These latter categories have certain
universality properties as monoidal categories, which consequently carry over
to $\widehat{\mathbf{T}}_{\mathfrak{gl}(V)}$,
$\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and
$\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$. Moreover, the categories
$\mathbb{T}_{\mathfrak{o}(V)}$ and $\mathbb{T}_{\mathfrak{sp}(V)}$ are known to
be equivalent, and this implies the equivalence of the categories
$\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and
$\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$. After introducing a supersymmetric
setting, we establish the equivalence of the category
$\widehat{\mathbf{T}}_{\mathfrak{gl}(V)}$ with the category
$\widehat{\mathbf{T}}_{\mathfrak{gl}(V\oplus \Pi V)}$, and the equivalence of
both categories $\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and
$\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$ with
$\widehat{\mathbf{T}}_{\mathfrak{osp}(V\oplus \Pi V)}$.
