I am faced with an operation $\otimes$ on a strict 2-category $C$ which walks and talks like a tensor, except that it only satisfies a "lax" interchange law. To be precise: for any $f,g,h,k\in C\_1$ with $f\circ h$ defined and $g\circ k$ defined, there is an "interchanger" 2-cell $I\_{f,g,h,k}:(f\otimes g)\circ(h\otimes k)\Rightarrow(f\circ h)\otimes(g\circ k)$.


Am I allowed to call such a thing a *lax tensor*? Would I need to show some coherence laws between interchangers? If yes, which?



Strict 2-Category with Lax Tensor?
This suggests that the monoidal product $\otimes\colon \mathbf{C}^2\to \mathbf{C}$ is a lax functor (see e.g. [here](https://kerodon.net/tag/008K) or [section 4.1 here](https://arxiv.org/abs/2002.06055)) instead of a strict 2-functor. If so, this determines which coherence laws you should expect. You don't mention identities, but assuming that they are preserved exactly this lax functor is strictly unitary, which simplifies some of the coherence conditions.


To warrant the symbol $\otimes$, you'd also expect your operation to be sufficiently associative and unital, and the resulting (pseudo?)naturality squares might also involve the interchangers.


