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does the given definition violate the principle of equivalence?
Yes, but see the article identity on objects functor for a discussion.
Add link to identity-on-objects functor.
- A functor $return : \mathcal{V}\to\mathcal{C}$ that is the identity on objects and preserves central morphisms.
“central morphisism” really needs to be defined here. I checked the Stanton & Levy paper, http://www.cs.ox.ac.uk/people/samuel.staton/papers/popl13.pdf where they define it.
If $t$ commutes with all $u$, then we say that $t$ is central. In other words, $t$ is central if it doesn’t matter when it is executed. A premulticategory is a multicategory if all morphisms are central. (Lambek [21] calls this the axiom of commutativity.)
This makes the nLab definition somewhat confusing. I guess it is just saying the $return$ functor maps all morphisms of $\mathcal{V}$ to central morphisms of $\mathcal{C}$.
( the referenced Lambeck paper can be found as the first reference on our Joachim Lambek page. )
( this distinction between multicategory and premulticategory seems to be the defining difference and really should be mentioned somewhere )
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