Let $F,G:C\to D$ be parallel functors such that there exists a natural isomorphism $\alpha:F\Rightarrow G$. Also, let $K:E\to C$ be a functor. Then there exists a natural isomorphism $\beta:FK\Rightarrow GK$.
Is this true? I think it should be. But I am unsure how to prove it.
Yes. Hint: We need to define a natural transformation $\beta$. So for every object $X$ in $E$, we need a map $\beta_X\colon F(K(X))\to G(K(X))$. Can you get such a map from $\alpha$? Now you just need to check that the maps $\beta_X$ cohere to a natural isomorphism, which will follow easily from the fact that $\alpha$ is a natural isomorphism.