  Let $A$ be a commutative noetherian ring and let $\mathfrak a$ be an ideal of
$A$. In this paper, we study the cofiniteness of an $A$- module via the
cofiniteness of its Koszul cohomology modules and vice versa. Moreover, we
introduce the category $\cK\cM({\bf x},\frak a,A)_{\rm cof}$ consisting of all
$A$-modules $M$ such that $H^i({\bf x},M)$ is $\frak a$-cofinite for all $i\geq
0$ and we study the abelianess of this new category.
