How to show that naturally isomorphic functors preserve the same limits? I've seen a number of times versions of the claim that if $F$ is naturally isomorphic to $G$, then if $F$ preserves limits of shape $\mathbf{J}$ so does $G$. (Sometimes with "It is easy to show that ...."!)
But my attempts to prove it have got into a tangle, so I must be missing some simple idea. I'd be very grateful then for (a sketch of) a proof!

D_S gives the headline idea. Perhaps the key thing to see is that -- in the terms of the proof below -- each face of the prism, not just the bottom face, is a naturality square. That gives you a lot to play with! So here's the proof spelt out rather more. And I've slightly edited things [from a version of notes of my own!] to fit the question:

