Prove that the composite of two continuous functors is continuous
I'm not sure I understand what this is asking.
A functor $H: C \rightarrow D$ is continuous if it preserves all small limits of a functor $F:J \rightarrow C$, which essentially means $H v: Ha \rightarrow HF$ is a limiting cone for all $v$ limiting cones of $F$ where $J$ is small.
But what I am specifically trying to prove in this? If $H$ preserves for $F$ and $H'$ preserves for $F'$ then am I trying to find an $F''$ that is preserved by $H'H$?
No. What you must prove is that if $H:C\to D$ and $H':D\to E$ are continuous, then so is $H'H:C\to E$. In fact you can prove something more precise : if $H$ and $H'$ preserve limits of all functors from a fixed small category $J$, then so does $H'H$ (so for example the composition of two functors that preserve products preserves products).