I don't know of any "clever" name, but there are names that appear in the literature, such as the tensor product of Grothendieck categories, or the category of (additive) locally presentable categories. Here are two references:

• Martin Brandenburg, Alexandru Chirvasitu, and Theo Johnson-Freyd, Reflexivity and dualizability in categorified linear algebra, Theory and Applications of Categories, Vol. 30, 2015, No. 23, pp 808-835. [link]. See Lemma 2.7.

• Julia Ramos González, On the tensor product of large categories, Thesis, University of Antwerp 2017. [link]. See section 2.4.

I don't know of any "clever" name, but there are names that appear in the literature, such as the tensor product of Grothendieck categories, or the category of (additive) locally presentable categories. Here are two references:

• Martin Brandenburg, Alexandru Chirvasitu, and Theo Johnson-Freyd, Reflexivity and dualizability in categorified linear algebra, Theory and Applications of Categories, Vol. 30, 2015, No. 23, pp 808-835. [link] See Lemma 2.7.

• Julia Ramos González, On the tensor product of large categories, Thesis, University of Antwerp 2017. [link] See section 2.4.