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.