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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?

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

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