# Does an adjoint of an internal Hom functor of a prounital closed category define a tensor product?

A closed category is a category equipped with internal Hom functors along with a unit object. Now this answer shows that if $$C$$ is a closed category whose internal Hom functor has a left adjoint, then this left adjoint defines a tensor product on $$C$$, i.e. it makes $$C$$ into a monoidal category. But I’m wondering if something more general is true.

A prounital closed category is like a closed category, except that the requirements regarding a unit object are dropped. (I don’t think this is standard terminology.) My question is, if $$C$$ is a prounital closed category whose internal Hom functor has a left adjoint, then does this left adjoint define some kind of tensor product on $$C$$? Now it presumably wouldn’t make $$C$$ monoidal, since you no longer have a unit object. But would it at least make $$C$$ into a “semigroupal category”, i.e. a category with a tensor product which satisfies all the properties of a monoidal category except the requirements regarding a unit object?

If not, does anyone know of a counterexample?

• Are there any examples you're interested in? – Qiaochu Yuan Mar 10 at 22:28
• @QiaochuYuan No, I was just curious about whether the result could be generalized. – Keshav Srinivasan Mar 10 at 22:31