Suppose that $X$ and $Y$ are two categories. Let $\operatorname{Funny}(X,Y)$ denote the category whose objects are functors $X\to Y$ and whose morphisms are *unnatural transformations* $F\to G$, where an unnatural transformation $F\to G$ is given by an $\operatorname{Ob}(X)$-indexed family of arrows $\gamma\_x:F(x)\to G(x)$, and that's it.


It is easy to see that its left adjoint gives a symmetric monoidal closed structure with unit object the terminal category. We call this monoidal product the funny tensor product and denote it by $\Box$. 


What I'd like to know is if the funny tensor product satisfies any algebraic "laws" with the ordinary cartesian product, by which I mean, if we have some kind of sequence of tensor products and cartesian products together with parentheses, for example, something like the sequence $$(((A\Box B)\times C)\Box D)\times E,$$ there is a way to either rearrange the parentheses or rewrite the expression with fewer or no parentheses. 


For instance, we can think of distributivity as an algebraic "law" between two operations. We can see associativity as an algebraic "law" between an operation and itself. However, I have never seen a nontrivial "law" between two associative operations having the same unit. Is there an extended form of the Eckmann-Hilton argument which shows that no nontrivial relationship of this type can ever be satisfied unless the two operations are equal (given that they are both associative and share a unit)?



Do the "funny" tensor product and the cartesian product satisfy any algebraic "laws"?
In a recent article



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> Mark Weber, *Free Products of Higher Operad Algebras*, Theory and Applications of Categories, Vol. 28, No. 2, 2013, pp. 24–65, [journal](http://www.tac.mta.ca/tac/volumes/28/2/28-02abs.html), arXiv:[0909.4722](https://arxiv.org/abs/0909.4722).
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there is a nice description of $□$ in terms of a pushout of the span $A\_0\times B\leftarrow A\_0\times B\_0\rightarrow A\times B\_0$. This works as well for the presheaf category of graphs (directed multigraphs, if you like). This pushout description yields canonical functors $A□(B\times C)\rightarrow(A□B)\times C$ that are easily seen not to be isomorphisms (both categories or graphs have the same objects, but different morphisms). In fact, $□$ as tensor and $\times$ as "par" provide a "linearly distributive structure" on both, the category of small categories and the category of small graphs. The latter notion is due to Robin Cockett and Robert Seely and was initially called "weak distributivity" (*Weakly distributive categories*, JPAA **114** Issue 2 (1997) pp 133--173, doi:[10.1016/0022-4049(95)00160-3](https://doi.org/10.1016/0022-4049%2995%2900160-3), plus [corrected version](http://www.math.mcgill.ca/rags/linear/wdc-fix.pdf) (pdf) on Robert Seely's web page).


