Trying to think a bit about an MO question*, and not being experienced in category theory, I happened to ask myself the following question that I'm quite sure would be pretty elementary for the non-layman in the field. Of course feel free to close in case it's too localized or anyway not suited for MO.
Given categories $\mathcal{A}$ and $\mathcal{B}$ and a functor $F:\mathcal{A}\to\mathcal{B}$, is it the case that every functor naturally equivalent to $F$ is of the form $K \circ F \circ H$ for $H$ an autoequivalence of $\mathcal{A}$ and $K$ an autoequivalence of $\mathcal{B}$ ?
*) [the one on the rigidity of the category of schemes, but it's not relevant here]