1. **State the problem:** Calculate the product of the mixed fraction $2 \frac{1}{3}$ and the fraction $\frac{4}{7}$.\n2. **Convert mixed fraction to improper fraction:** $2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$.\n3. **Multiply the fractions:** $\frac{7}{3} \times \frac{4}{7}$.\n4. **Simplify before multiplying:** Cancel the common factor 7 in numerator and denominator: $\frac{\cancel{7}}{3} \times \frac{4}{\cancel{7}} = \frac{1}{3} \times \frac{4}{1}$.\n5. **Multiply numerators and denominators:** $\frac{1 \times 4}{3 \times 1} = \frac{4}{3}$.\n6. **Convert improper fraction to mixed number:** $\frac{4}{3} = 1 \frac{1}{3}$.\n\n**Final answer:** $1 \frac{1}{3}$