Let $F:A\rightarrow B$ be an equivalence between categories and $H:B\rightarrow C$ be a limit creating functor.
Prove $HF$ is a limit reflecting functor.
I am stumped. I know I have to use the fact that $F$ is an equivalence between categories and use all of the extra structure that comes with such a functor, but I'm not really sure how.
$H$ reflects limits - that's one part of the definition of creating them. $F$ preserves and reflects everything "categorical", including limits†. Finally, limit-reflecting functors are closed under composition.
Can you prove/follow the claims above?
†in fact, all fully faithful functors reflect limits
Edit: Here's the proof of the stronger statement in the footnote:
Let $F : A → B$ be fully faithful, $D : I → A$ a diagram, and $α : ΔL ⇒ D$ a cone such that $Fα$ is a limiting cone. We need to prove that $α$ is limiting itself, or in other words every other cone $β : ΔK ⇒ D$ needs to factor through $α$ via a unique morphism $φ : K → L$.
Since $F$ is fully faithful, the morphisms $K → L$ correspond exactly to morphisms $FK → FL$, which is a good thing because you know nothing about the former, and something about the latter. In particular, you know there exists a unique morphism $ψ : FK → FL$ which factors $Fβ$ through the limiting cone $Fα$ (ie. $Fβ_i = Fα_i ∘ ψ$, for all $i$ in $I$). Now let $φ$ be the unique morphism such that $ψ = Fφ$, and check that it really proves that $α$ is limiting. Tell me if you need me to fill in any other details.