To solve the expression \((-2) \times \left(\frac{1}{5}\right) \div \left(\frac{2}{3}\right) \times \left(-\frac{5}{6}\right) \div \left(\frac{3}{4}\right)\), we will simplify it step by step by applying the properties of operations.
First, we can rewrite the division as multiplication by the reciprocal:
\[ (-2) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(-\frac{5}{6}\right) \times \left(\frac{4}{3}\right) \]
Next, let's combine the fractions:
\[ (-2) \times \left(\frac{1}{5}\right) \times \frac{3}{2} \times (-\frac{5}{6}) \times \frac{4}{3} \]
Now let's group it together:
\[ = \frac{(-2) \times 1 \times 3 \times (-5) \times 4}{5 \times 2 \times 6 \times 3} \]
Now calculate the numerator:
\[ (-2) \times (-5) = 10 \] \[ 10 \times 1 \times 3 \times 4 = 120 \]
Now calculate the denominator:
\[ 5 \times 2 = 10 \] \[ 10 \times 6 = 60 \] \[ 60 \times 3 = 180 \]
So we have:
\[ \frac{120}{180} \]
We can simplify \(\frac{120}{180}\):
\[ 120 \div 60 = 2 \] \[ 180 \div 60 = 3 \]
Thus,
\[ \frac{120}{180} = \frac{2}{3} \]
So, the answer is:
\[ \boxed{\frac{2}{3}} \]