To simplify the expression $\left(\frac{1}{x^{-1}} \div \frac{1}{y^{-1}}\right)^{-1}$, first rewrite the division as multiplication by the reciprocal, resulting in $\frac{y^{-1}}{x^{-1}}$. This simplifies to $\frac{y}{x}$ when the negative exponents are converted to positive. Finally, applying the outer exponent of $-1$ gives the final result of $\frac{y}{x}$.