Let $\mathcal{C}$ be a filtered category and $\mathcal{D}$ a finite category. Consider a functor $F:\mathcal{C}\times\mathcal{D}\rightarrow\textbf{Ab}$. I want to show that $$\text{co}\!\lim_C\lim_D F(C,D)\cong\lim_D\text{co}\!\lim_C F(C,D).\tag{1}$$
Let $U:\textbf{Ab}\rightarrow\textbf{Set}$ be the forgetful functor. Then $U\circ F:\mathcal{C}\times\mathcal{D}\rightarrow\textbf{Set}$. Thus $$\text{co}\!\lim_C\lim_D U(F(C,D))\cong\lim_D\text{co}\!\lim_C U(F(C,D)).\tag{2}$$ How can I use (2) along with the fact that $U$ reflects and preserves filtered colimits in order to conclude (1)? What are the steps involved in such an argument? (E.g. are there "algebraic" manipulations that take us from (2) to (1)?)
Edit:
Can I just use the fact that $U$ reflects isomorphisms?