Deligne categories and the limit of categories $Rep(GL(m\mathrm{\mid }n))$

For each integer $t$ a tensor category $V_t$ is constructed, such that exact tensor functors $V_t⟶C$ classify dualizable $t$-dimensional objects in $C$ not annihilated by any Schur functor. This means that $V_t$ is the “abelian envelope” of the Deligne category $Rep(G{L}_t)$. Any tensor functor $Rep(G{L}_t)⟶C$ is proved to factor either through $V_t$ or through one of the classical categories $Rep(GL(m\mathrm{\mid }n))$ with $m-n=t$. The universal property of $V_t$ implies that it is equivalent to the categories $Re{p}_{Rep(G{L}_{t_1})\otimes Rep(G{L}_{t_2})}(GL(X),ϵ)$, ($t=t_1+t_2$, $t_1$ not integer) suggested by Deligne as candidates for the role of abelian envelope.