Let $K$ be an algebraically closed field. On page 31 of the book Elements of representation theory of associative algebras, volume 1, from Theorem 5.13 (a) we see that $D=\hom_K(\cdot, K)$ is exact. But we know that $\hom$ is left exact and not necessarily exact. I am confused about this. Is it because $D=\hom_K(\cdot, K)$ but not $\hom_A(\cdot, A)$, where $A$ is a $K$-algebra. Thank you very much.