I am interested in finding a definition of monoidal category that ultimately relies only on category theory elements such as objects, arrows, and categories. For instance, it should never assume we are talking about small categories because that relies on what a set is.
From nLab, I get that the tensor product is a functor. Then, for the definition of functor, there are two options. The first options (external definition) says that a functor is a mapping satisfying some properties. Here, "mapping" seems to be relying on sets already, so I think I need to use the second definition.
The second definition (internal definition) says that a functor is a pair of morphisms in a category for which its target and source categories are internal categories. Finally, an internal category is defined as 2 objects and 4 morphisms from the ambient category satisfying a few properties.
The problem is that I don't know how to put together this second definition of functor with what appears in the definition of monoidal category.
The functor appearing the definition of monoidal category is $\otimes: C \times C \to C$, but since, in the definition using internal categories, I never see the category product $\times$ (which is to be expected, since internal categories don't appear inside the ambient category as single objects), I don't know how to put the definitions of functor and of monoidal category together.
Is such a definition even possible? I appreciate any help with this.
PS: Trying to answer this, I see in a few places mentions to 2-categories, but for these I also don't see a definition in terms only of category.