Let $f: A \to B$ and $g: B \to C$ be uniformly continuous. What can be said about $g\circ f: A \to C$?
We have only briefly covered uniform continuity so it is unclear what I should see about $g\circ f$.
I guess A, B and C are metric spaces. Fix $\varepsilon>0$. Then use the uniform continuity of $g$ on $B$: you will find a $\delta$ such that $|g(x)-g(y)|\leq\varepsilon$ if $d_B(x,y)\leq \delta$. Now, can you find a $\delta'$ such that if $a,a' \in A$ and $d_A(a,a')\leq \delta'$ then $|g \circ f(a)-g \circ f(a')|\leq \varepsilon$?