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I've read that $Rel$ (the category of sets and relations) is a 2-category by considering 2-morphisms to be inclusion of relations. Is $Set$ also a 2-category by considering 2-morphisms to be inclusion of functions (as relations)? I can't find the reason why this is not true.

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    $\begingroup$ Maybe you want to consider the partial functions as morphisms between sets? $\endgroup$ – Berci May 31 '16 at 20:05
  • $\begingroup$ @Berci In that case we have a 2-category again? $\endgroup$ – user343618 May 31 '16 at 21:36
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Give two sets $A,B$, and $f,g:A\to B$, if $f\subseteq g$ as relations, then $f=g$.

So this is a trivial 2-category - the only 2-maps are the identity maps - but it is a 2-category.

So it is a $2$-category in exactly the same way that any category can be made a $2$-category.

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