functional analysis - Prove sets of continuous mappings are the same

Let $C([0,T];C(\overline{U}))$ denote the set of all continuous functions $u:[0,T]\rightarrow C(\overline{U})$ with
$$\|u\|_{C([0,T];C(\overline{U}))}:=\max_{0\leqslant t \leqslant T} \|u(t)\|<\infty$$

Prove that $C([0,T];C(\overline{U}))=C([0,T]\times \overline{U})$

I am skeptical this is even true. I feel like we could apply a theorem from topology regarding the product space, but am having little success. Not really sure how to approach such a problem. Are there any counter examples that disprove the above? Any help would be much appreciated.