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I am currently stuck at the following problem, and I was hoping that some might know of some literature or known results that might enable me to tackle it.

Let $F,G:\mathcal{C}\rightarrow \mathsf{Sets}$ be 'nice' functors from a 'nice' category $\mathcal{C}$ (i.e. (co)limits exist and are preserved). Assume further that the monoids $\mathsf{End}(F)$ and $\mathsf{End}(G)$ are isomorphism (as monoids).

Does the latter help me at all in showing that $F$ and $G$ are isomorphic themselves? Are there any other 'sensible' conditions we could require for this to be the case?

If natural endomorphisms being isomorphic meant that for all $A\in \mathcal{C}$, $\mathsf{End}(F(A))$ were isomorphic to $\mathsf{End}(G(A))$, I think this might have been true in $\mathsf{Sets}$ at least. But that is not quite what endofunctors are.

Anyone have any ideas? Cheers.

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    $\begingroup$ If $\mathcal{C}$ has an automorphism $T$, and $G=F\circ T$, then the endomorphism monoids of $F$ and $G$ will be isomorphic, but there's no reason to expect $F$ and $G$ to be. $\endgroup$
    – Jeremy Rickard
    Commented Aug 3, 2018 at 10:34
  • $\begingroup$ Thank you very much. This seems to be true even in the "best of cases". I will have to think of something else. Cheers. Edit: Not sure if I should delete this thread... prob. not, but if mods are OK with it, feel free to do so. $\endgroup$
    – I.P
    Commented Aug 3, 2018 at 11:42

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