1. The problem is to evaluate the expression $4\frac{1}{3} - 5\frac{1}{5} + \frac{4}{s} \text{ of } \frac{2}{3}$.\n\n2. Convert mixed numbers to improper fractions:\n$4\frac{1}{3} = \frac{13}{3}$\n$5\frac{1}{5} = \frac{26}{5}$\n\n3. The expression becomes:\n$$\frac{13}{3} - \frac{26}{5} + \frac{4}{s} \times \frac{2}{3}$$\n\n4. To combine the first two fractions, find a common denominator, which is 15:\n$$\frac{13}{3} = \frac{13 \times 5}{3 \times 5} = \frac{65}{15}$$\n$$\frac{26}{5} = \frac{26 \times 3}{5 \times 3} = \frac{78}{15}$$\n\n5. Subtract the fractions:\n$$\frac{65}{15} - \frac{78}{15} = \frac{65 - 78}{15} = \frac{-13}{15}$$\n\n6. Now add the last term:\n$$\frac{-13}{15} + \frac{4}{s} \times \frac{2}{3} = \frac{-13}{15} + \frac{8}{3s}$$\n\n7. The final simplified expression is:\n$$\boxed{\frac{-13}{15} + \frac{8}{3s}}$$\n\nNote: Without a specific value for $s$, this is the simplest form.