Dynamic Programming - Policy Iteration

I just completed the Optimal Growth II lecture. I was wondering if there was a way to extract the value function from the policy function. As far as I can see, there seems to be no elegant way to do this.

I get that what we care about might be just the policy function anyway, but just thought it would be useful to derive the value function.

There is, actually. See the solution to exercise 2, where an integral operation recovers the value function.