which admits a right adjoint. This right adjoint is precisely the bifunctor , once we noticed that the category comes naturally equipped with an arrow induced by (bi)functoriality of , starting from the canonical arrows to the terminal category.
It is quite clear that is defined by sending to the pair of categories . The bijection
is now rather obvious, since any functor determines a functor and viceversa.
If the small categories are two posets, their join consists of their ordinal sum;
The monoidal structure induced on by is not symmetric (if it was, then the right cone of would be equivalent to the left cone, which is blatantly false);
The monoidal structure induced on by is not closed, since the functor does not preserve colimits.
The functor is a left adjoint, and similarly is the functor ; see Joyal, Ch. 3.1.1-2 for a detailed description.
The cone below a category is the join . The cone above is the join .