  Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, $M$ be a
finitely generated $R$-module and $\mathfrak{a}$, $I$ and $J$ be ideals of $R$.
We investigate the structure of formal local cohomology modules of
$\mathfrak{F}^i_{\mathfrak{a},I,J}(M)$ and
$\check{\mathfrak{F}}^i_{\mathfrak{a},I,J}(M)$ with respect to a pair of
ideals, for all $i\geq 0$. The main subject of the paper is to study the
finiteness properties and Artinianness of
$\mathfrak{F}^i_{\mathfrak{a},I,J}(M)$ and
$\check{\mathfrak{F}}^i_{\mathfrak{a},\mathfrak{m},J}(M)$. We study the maximum
and minimum integer $i\in \N$ such that
$\mathfrak{F}^i_{\mathfrak{a},\mathfrak{m},J}(M)$ and
$\check{\mathfrak{F}}^i_{\mathfrak{a},\mathfrak{m},J}(M)$ are not Artinian. We
obtain some results involving cossuport, coassociated and attached primes for
formal local cohomology modules with respect to a pair of ideals. Also, we give
an criterion involving the concepts of finiteness and vanishing of formal local
cohomology modules and \v{C}ech-formal local cohomology modules with respect to
a pair of ideals.
