Definition 0.1   A quantum category $\Q$ is defined as the (non-Abelian) category of quantum groupoids, , and quantum groupoid homomorphisms, , where $i$ and $j$ are indices in an index class, $\mathbf{I}$ , all subject to the usual ETAC axioms and their interpretations.

Remark 0.1   The category of quantum groupoids, , is trivially a subcategory of the groupoid category, that can also be regarded as a functor category, or $2$ -category, if is small, that is, if $G^0$ is a set rather than a class.

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Baianu, I.C.: 1971a, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science, September 1-4, 1971, the University of Bucharest.

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