  Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ an
arbitrary $R$-module and $N$ a finite $R$-module. We prove that \cite[Theorem
2.1]{Mel} and \cite[Proposition 3.3 (i)$\Leftrightarrow$(ii)]{B1} are true for
any Serre subcategory of $R$-modules. We also prove a characterization theorem
for $\lc_{\fa}^{i}(M)$ and $\lc_{\fa}^{i}(N,M)$ to be $\fa$-cofinite for all
$i$, whenever one of the following cases holds: (a) $\ara (\fa)\leq 1$, (b)
$\dim R/\fa \leq 1$ or (c) $\dim R\leq 2$. In the end we study Artinianness and
Artinian $\fa$-cofiniteness of local cohomology modules.
