  We extend previous work by constructing a universal abelian tensor category
${\bf T}_t$ generated by two objects $X,Y$ equipped with finite filtrations
$0\subsetneq X_0\subsetneq ... X_{t+1}\subsetneq X$ and $0\subsetneq
Y_0\subsetneq ... Y_{t+1}\subsetneq Y$, and with a pairing $X\otimes Y\to
\mathbb{I}$, where $\mathbb{I}$ is the monoidal unit. This category is modeled
as a category of representations of a Mackey Lie algebra
$\mathfrak{gl}^M(V,V_*)$ of cardinality $2^{\aleph_t}$, associated to a
diagonalizable pairing between two complex vector spaces $V,V_*$ of dimension
$\aleph_t$. As a preliminary step, we study a tensor category $\mathbb{T}_t$
generated by the algebraic duals $V^*$, $(V_*)^*$. The injective hull of
$\mathbb{C}$ in $\mathbb{T}_t$ is a commutative algebra $I$, and the category
${\bf T}_t$ is consists of the free $I$-modules in $\mathbb{T}_t$. An essential
novelty in our work is the explicit computation of Ext-groups between simples
in both categories ${\bf T}_t$ and $\mathbb{T}_t$, which had been an open
problem already for $t=0$. This provides a direct link from the theory of
universal tensor categories to Littlewood-Richardson-type combinatorics.
