Let $k$ be a field and $\mathcal{T}$ be a $k$-linear triangulated category with finite dimensional spaces of morphisms. Bondal and Kapranov proved that every Serre functor on $\mathcal{T}$ is exact (Proposition 3.3 [here][1]). A different proof was given by Huybrechts in his Fourier-Mukai book (Proposition 1.46). The proof by Bondal and Kapranov is a bit technical and relies on the notion of a mattress. For now, I decided not to read it thoroughly. Concerning the one given by Huybrechts, I could not convince myself that it is complete. Could someone please shed a light on this, possibly providing a simple proof or explaining how the desired linear form in Huybrechts' proof is constructed?

  [1]: https://iopscience.iop.org/article/10.1070/IM1990v035n03ABEH000716