Let's go back to the issue with condition of tensor distributing over joins for the existence of internal homs.

It seems that Matthew is saying that it is not necessary that a monoidal preorder has the monoidal product distributing over joins for it to be closed.

On the other hand he is saying that by the adjoint functor theorem he can show that a certain monoidal preorder is not closed because the monoidal product does not distribute over the joins.

These two things *seem* to contradict each other. Am I misunderstanding the positions here?