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Let $T_1, T_2$ be two semisimple tensor categories over a field $k$ (i.e. symmetric rigid monoidal abelian $k$-linear). Then is the Deligne product, $T_1\otimes T_2$ also a $k$-tensor category?

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Yes, it is. In the semi-simple case Deligne product is very explicit, see for example BaKi Definition 1.1.15

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