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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?

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This suggests that the monoidal product $\otimes\colon \mathbf{C}^2\to \mathbf{C}$ is a lax functor (see e.g. here or section 4.1 here) 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.

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    $\begingroup$ However, formulating the associativity and unit conditions may be tricky, because lax functors do not form a tricategory. You may want to ask whether your 2-category happens to actually be a double category. $\endgroup$
    – Mike Shulman
    Commented May 17, 2021 at 19:16
  • $\begingroup$ Thank you, it did help to think of this thing as a lax functor from $C\times C$ to $C$. My concrete $\otimes$ is indeed strictly unital and also well-typed and associative. On the 1-restriction of $C$, I'd call it a pretensor; but I'm being told not to... $\endgroup$
    – Uli Fahrenberg
    Commented Jun 13, 2021 at 10:02

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