In this PDF, page 5 says:
Given a set of functions $f(x_i,pa(x_i))$ non-negative and sum to 1, we define a joint probability as follow: $$p(x_1,\dots, x_n)=\prod_i f(x_i,pa(x_i))$$ [...] what is the relationship between the conditional probability $p(x_i|pa(x_i))$ and $f(x_i,pa(x_i))$, a function with has the properties of a conditional function but it is arbitrary? As will see, this function is in the fact one and same. That is, under the definition of joint probability, the function $f(x_i,pa(x_i))$ is $p(x_i|pa(x_i))$.
How can it go from $f(x_i,pa(x_i))$ to $p(x_i|pa(x_i))$ and state that it is the only possible function that can be assigned to $f(x_i,pa(x_i))$?