The Gray tensor product is a “better” replacement for the cartesian product of strict 2-categories. To get the idea it suffices to consider the 2-category $\mathbf{2}$ which has two objects, 0 and 1, one non-identity morphism $0\to 1$, and no nonidentity 2-cells. Then the cartesian product $\mathbf{2}\times\mathbf{2}$ is a commuting square, while the Gray tensor product $\mathbf{2}\otimes\mathbf{2}$ is a square which commutes up to isomorphism.
More generally, for any 2-categories $C$ and $D$, a 2-functor $C\times\mathbf{2} \to D$ consists of two 2-functors $C\to D$ and a strict 2-natural transformation between them, while a 2-functor $C\otimes\mathbf{2} \to D$ consists of two 2-functors $C\to D$ and a pseudonatural transformation between them.
Following up on the last comment, $B\otimes C$ can be defined by
where $Ps(C,D)$ is the 2-category of 2-functors, pseudonatural transformations, and modifications $C\to D$. In other words, the category $2Cat$ of strict 2-categories and strict 2-functors is a closed symmetric monoidal category, whose tensor product is $\otimes$ and whose internal hom is $Ps(-,-)$.
When considered with this monoidal structure, $2Cat$ is often called $Gray$. Gray-categories, or categories enriched over $Gray$, are a model for semi-strict 3-categories. Categories enriched over $2Cat$ with its cartesian product are strict 3-categories, which are not as useful. This is one precise sense in which the Gray tensor product is “more correct” than the cartesian product.
$Gray$ is a rare example of a non-cartesian monoidal category whose unit object is nevertheless the terminal object — that is, a semicartesian monoidal category.
There are also versions of the Gray tensor product in which pseudonatural transformations are replaced by lax or oplax ones. (In fact, these were the ones originally defined by Gray.)
$Gray$ is actually a monoidal model category (that is, a model category with a monoidal structure that interacts well with the model structure), which $2Cat$ with the cartesian product is not. In particular, the cartesian product of two cofibrant 2-categories need not be cofibrant. This is another precise sense in which the Gray tensor product is “more correct” than the cartesian product.
The cartesian monoidal structure is sometimes called the “black” product, since the square $2\times 2$ is “completely filled in” (i.e. it commutes). There is another “white” tensor product in which the square $2\Box 2$ is “not filled in at all” (doesn’t commute at all), and the “gray” tensor product is in between the two (the square commutes up to an isomorphism). This is a pun on the name of John Gray, for whom the Gray tensor product is named. The “white” tensor product is also called the funny tensor product.
There are generalizations to higher categories of the Gray tensor product. In particular there is a tensor product on strict omega-categories – the Crans-Gray tensor product – which is such that restricted to strict 2-categories it reproduces the Gray tensor product.
A closed monoidal structure on strict omega-categories is introduced by Al-Agl, Brown and Steiner. This uses an equivalence between the categories of strict (globular) omega categories and of strict cubical omega categories with connections; the construction of the closed monoidal structure on the latter category is direct and generalises that for strict cubical omega groupoids with connections established by Brown and Higgins.
John W. Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics, Vol. 391. Springer-Verlag, Berlin-New York, 1974. xii+282 pp doi:10.1007/BFb0061280 (see also Adjointness for 2-Categories)
John W. Gray, Coherence for the Tensor Product of 2-Categories, and Braid Groups , pp.62-76 in Heller, Tierney (eds.), Algebra, Topology, and Category Theory , Academic Press New York 1976.
Robert Gordon, John Power, Ross Street. Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), no. 558, vi+81 pp. doi:10.1090/memo/0558 (AMS bookstore incl. free sample chapter)
Stephen Lack, A Quillen model structure for 2-categories, K-Theory, 26(2) (2002) pp171-205, (gzipped .ps) (doi:10.1023/A:1020305604826 - requires Portico subscription)
Stephen Lack, A Quillen model structure for bicategories, K-theory, 33(3) (2004) pp185-197, (gzipped .ps) (doi:10.1007/s10977-004-6757-9 - requires Portico subscription)
Ronnie Brown and P.J. Higgins, Tensor products and homotopies for $\omega$-groupoids and crossed complexes, J. Pure Appl. Alg. 47 (1987) 1-33. doi:10.1016/0022-4049(87)90099-5, (pdf)
F.A. Al-Agl, R. Brown and R. Steiner, Multiple categories: the equivalence between a globular and cubical approach, Advances in Mathematics, 170 (2002) 71-118. doi:10.1006/aima.2001.2069, arXiv:math/0007009
The Gray tensor product as the left Kan extension of a tensor product on the full subcategory $Cu$ of $2Cat$ is on page 16 of
A general theory of lax tensor products, unifying Gray tensor products with the Crans-Gray tensor product is in
A proof that the Gray tensor product does form a monoidal structure, based only on its universal property, is in
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