Dear all, In "Formal Category Theory: Adjointness for 2-Categories", Gray introduces what is now known as the Gray tensor product. The following two questions concern the part I,4.23, Appendix A, p.92-95, where Gray relates the aforesaid product to a universal construction due to Bénabou. (I shall send a PDF version of what follows for those averse to e-mails swarming with dollar symbols.) In the following, all the 2-categories are strict. 1. Given a 2-category $A$, Gray, refering to Bénabou unpublished notes, gives a description of a couple $(\widetilde{A}, P)$ where $ \widetilde{A}$ is a 2-category and $P : A \to \widetilde{A}$ a lax functor (oplax in Gray's text, but I take the liberty to take the lax viewpoint and translate everything accordingly) which has the following universal property: for any 2-category $B$ and any lax 2- functor $F : A \to B$, there is a unique strict 2-functor $\widetilde {F} : \widetilde{A} \to B$ such that $\widetilde{F}P=F$. I have trouble in understanding the description with which Gray goes on, but I believe it is equivalent to the following one, which I have learnt from Matias del Hoyo: The objects of $\widetilde{A}$ are those of $A$; If $a$ and $a'$ are two objects of $\widetilde{A}$, then the 1-cells from $a$ to $a'$ in $\widetilde{A}$ are the couples $([m], (x) : [m] \to A)$, where $m$ is a nonnegative integer, $[m]$ denotes the category associated to the ordinal $m$, and $(x) : [m] \to A$ is a 2- functor such that $(x)(0)=a$ and $(x)(1)=a'$, in other words $(x)$ can be viewed as a chain linking $a$ to $a'$ of 1-cells of $A$ (we do not require the chain to be nondegenerate, so identities are allowed in it); If $a$ and $a'$ are two objects of $\widetilde{A}$ and $([m], (x) : [m] \to A)$ and $([n], (y) : [n] \to A)$ are two 1-cells from $a$ to $a'$ in $\widetilde{A}$, then the 2-cells from $([m], (x))$ to $([n], (y))$ in $\widetilde{A}$ are the n+1-uples $(\varphi, \alpha_{1}, ldots, \alpha_{n})$, where $\varphi : [n] \to [m]$ is a morphism between partially ordered sets (we do not require it to be injective) such that, for every $i$ in $[n]$, we have $x_{\varphi(i)} = y_{i}$, and where $\alpha_{i}$ is a 2-cell of $A$ from $x_{\varphi(i), \varphi (i)-1} \circ \ldots \circ x_{\varphi(i-1)+1, \varphi(i-1)}$ to $y_{i, i-1}$ for $i \in \{1, \ldots, n\}$. ($x_{i+1,i}$ stands for the image of the arrow $i \to i+1$ by $(x)$.) The first question I have is: does this explicit description appear in the literature somewhere else than Gray's book? Is there a detailed proof of the universal property written down to which I could refer? I confess having too much difficulties in understanding Gray's book not to hope that someone will point out a more concrete and modern treatment in the literature. (In fact, I even have trouble in convicing myself that the description just given is equivalent to that given by Gray, one of the reason being the terminological discrepancy between Gray's words and current usage.) 2. Gray goes on by relating the Gray tensor product to this universal construction. Given two 2-categories $A$ and $B$, I would like to understand the relationship between the 2-category $\widetilde{A \times B}$ ("tilde construction" of the cartesian product of $A$ and $B$) and the Gray tensor product $A \otimes B$ of $A$ and $B$. I suspect there is some kind of "localization" 2-functor relating these two 2-categories. What is the more concrete description of this canonical 2-functor? Once again, pointers to the literature are welcome. I would like to stress that I am not at all interested in restricting the description to the pseudo case of the Gray tensor product, I really want to deal with the general lax case. Thanks in advance! Jonathan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]