I am a novice in category theory, and I am studying the notion of Pro category. I wonder if the set $\operatorname{Hom}(0,0)$ only contains the identity morphism.

If it does not, since (I guess) any two morphisms in $\mathrm{Hom}(0,0)$ commute with respect to both the tensor product and the composition, as a consequence of the Eckmann-Hilton argument, is this the case with other morphisms ? More precisely, does a morphism $\mu$ in $\mathrm{Hom}(0,0)$ commute with a morphism $\mu'$ in $\mathrm{Hom}(n,m)$ with respect to the tensor product, i.e. $\mu\otimes\mu' = \mu' \otimes \mu$ ?

Moreover, is this the case for the the endomorphisms associated with the unit of any monoidal category?

Thank you all by advance !

  • $\begingroup$ I guess, $\mu$ would not necessarily commute with $\mu'$. $\endgroup$ – Berci Oct 16 '16 at 23:54
  • $\begingroup$ Thank you @Berci. Do you have any idea of the proof ? And do you have any guess about the morphisms in the set $\mathrm{Hom}(0,0)$ ? $\endgroup$ – Ludovic Mignot Oct 22 '16 at 9:21

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