Motivated by the answer to this question, I will ask the following question: Let $\mathcal{A}$ and $\mathcal{B}$ be small semisimple abelian categories. Let $U:\mathcal{A} \to \mathcal{B}$ be a functor that preserves and reflects exact sequences.

Is this enough in general to give an adjoint functor such that the unit of the adjunction is injective? If not, then what are examples of assumptions we can add to make this true?