To evaluate $(f \cdot g)(2)$ for the functions $f(x) = x - 5$ and $g(x) = x^2 - 2x + 3$, first use the product function $(f \cdot g)(x) = x^3 - 7x^2 + 13x - 15$. Substitute $x = 2$ into this polynomial: $(f \cdot g)(2) = 2^3 - 7(2)^2 + 13(2) - 15$. Simplifying the expression yields $8 - 28 + 26 - 15$, which sums to $-9$. Therefore, $(f \cdot g)(2) = -9$.