Vertex algebras and $2$-monoidal categories

In this article we address Problem 5.12 in $\href{https://projecteuclid.org/ebook/Download?urlId=10.2969/aspm/03110051&isFullBook=false}{[9]}$. More precisely, we prove that the singular tensor product introduced by R. Borcherds in the previous reference is part of a 2-monoidal category structure in a certain category of functors. We also complete some missing points in the previously mentioned article, most notably in the definitions of singular tensor products and of vertex algebras themselves, which are however verified in all the examples appearing in that reference. To prove our results it will be extremely useful, if not essential, to frame our objects within the language of bicategories. We also introduce a slightly more general notion of (quantum) vertex algebra than the one in $\href{https://projecteuclid.org/ebook/Download?urlId=10.2969/aspm/03110051&isFullBook=false}{[9]}$, that we call categorical (quantum) vertex algebra, enjoying all the properties mentioned by Borcherds in that article and having as particular example the definition presented by that author.