I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev. In section 2.3 "Fiber functors and reconstruction theorems*, the following definition is given for a $C^*$-tensor category $\mathscr{C}$:

Definition: A tensor functor $F: \mathscr{C}\to \text{Hilb}_f$ is called a fiber functor if it is faithful and exact.

Question: What does exactness of a functor mean in this context? I know what it means in the context of (semi)abelian categories where it means that the functor $F$ preserves short exact sequences, but I believe in a general $C^*$-tensor category (without dual objects) it is possible that kernels and cokernels do not exist, so the aforementioned notion of exactness doesn't seem to apply in this situation.

I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset. In section 2.3 "Fiber functors and reconstruction theorems*, the following definition is given for a $C^*$-tensor category $\mathscr{C}$:

Definition: A tensor functor $F: \mathscr{C}\to \text{Hilb}_f$ is called a fiber functor if it is faithful and exact.

Question: What does exactness of a functor mean in this context? I know what it means in the context of (semi)abelian categories where it means that the functor $F$ preserves short exact sequences, but I believe in a general $C^*$-tensor category (without dual objects) it is possible that kernels and cokernels do not exist, so the aforementioned notion of exactness doesn't seem to apply in this situation.