I'm trying to prove the following fact:
Let $F, G : \mathcal{C} \to \mathcal{D}$ be functors and let $\alpha : F \to G$ be a cartesian natural transformation i.e. a transformation such that all naturality squares of $\alpha$ are pullback squares. If $G$ preserves colimits of a certain shape, then $F$ does also.
Let $J : I \to \mathcal{C}$ be a diagram. I figure that we have a canonical diagram
$ \require{AMScd} \begin{CD} \text{colim } FJ @>>> F(\text{colim } J)\\ @VVV @VVV \\ \text{colim } GJ @>>> G(\text{colim } J) \end{CD} $
where the bottom arrow is an isomorphism. I'm not sure where to proceed from here, or how to use the cartesianness of $\alpha$. How can I prove this?