  The duality of finitary biframes as pointfree bitopological spaces is
explored. In particular, for a finitary biframe $\mathcal{L}$ the ordered
collection of all its pointfree bisubspaces (i.e. its biquotients) is studied.
It is shown that this collection is bitopological in three meaningful ways. In
particular it is shown that, apart from the assembly $\mathsf{A}(\mathcal{L})$
of a finitary biframe $\mathcal{L}$, there are two other structures
$\mathsf{A}_{cf}(\mathcal{L})$ and $\mathsf{A}_{\pm}(\mathcal{L})$, which both
have the same main component as $\mathsf{A}(\mathcal{L})$. The main component
of both $\mathsf{A}_{cf}(\mathcal{L})$ and $\mathsf{A}_{\pm}(\mathcal{L})$ is
the ordered collection of all biquotients of $\mathcal{L}$. The structure
$\mathsf{A}_{cf}(\mathcal{L})$ being a biframe shows that the collection of all
biquotients is generated by the frame of the patch-closed biquotients together
with that of the patch-fitted ones. The structure
$\mathsf{A}_{\pm}(\mathcal{L})$ being a biframe shows the collection of all
biquotients is generated by the frame of the positive biquotients together with
that of the negative ones. Notions of fitness and subfitness for finitary
biframes are introduced, and it is shown that the analogues of both
characterization theorems for these axioms appearing in Picado and Pultr (2011)
hold. A spatial, bitopological version of these theorems is proven, in which
finitary biframes whose spectrum is pairwise $T_1$ are characterized, among
other things in terms of the spectrum of $\mathsf{A}_{cf}(\mathcal{L})$.
