Now we’re going to assume that our monoidal category is also cocomplete. In particular, we’ll assume that it has all small coproducts.
This is enough to ensure that the “underlying set” functor has a left adjoint that sends a set to the coproduct of a bunch of copies of indexed by . The adjunction says that is naturally isomorphic to . That is, a function from to the underlying set of is the same as an -indexed collection of elements of the underlying set of .
It’s straightforward from here to verify that this adjunction interchanges the cartesian product on and the monoidal structure on . That is, and .
And now the 2-functor has a left 2-adjoint . Starting with an ordinary category (with hom-sets) we get the “free -category” with the same objects as , and with the hom-objects given by . Compositions and identities for this -category are induced by the above exchange of cartesian and monoidal structures. The actions of this 2-functor on functors and natural transformations are straightforwardly defined.
For example, when , we just replace each hom-set by the free abelian group on . We extend the composition and identity maps from the ordinary category by linearity. In particular, if has only one object — if it’s a monoid — then is the free ring on .
To finish off, let be a small category, so is a small -category. Then we can define functor categories and functor -categories. Verify that by the above 2-adjunction.