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

On factorizing $F$-matrices in $Y(sl_n)$ and $U_q(\hat{sl_n})$ spin chains