  We consider quantum spin chains arising from $N$-fold tensor products of the
fundamental evaluation representations of $Y(sl_n)$ and $U_q(\hat{sl_n})$.
Using the partial $F$-matrix formalism from the seminal work of Maillet and
Sanchez de Santos, we derive a completely factorized expression for the
$F$-matrix of such models and prove its equivalence to the expression obtained
by Albert, Boos, Flume and Ruhlig. A new relation between the $F$-matrices and
the Bethe eigenvectors of these spin chains is given.
