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?

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?

Edit: In response to the comments below, the definition of semisimple is the one taken from nLab, so yes I am assuming finite direct sums.

An abelian category is called semisimple if every object is a semisimple object, hence a direct sum of finitely many simple objects.

Edit: I am happy to assume that my categories are linear, that is enriced over vector spaces.