# How can i show that the the category of pointed sets and the category of sets and partial functions are not isomorphic?

Denote $Pf$ the category of sets and partial functions and $Set_*$ the category of pointed sets.

i can't see why, i was trying some contradiction argument with the definition of isomorphism, maybe taking some particular sets.... supose exists functors $F: Pf \rightarrow Set_*$ and $G: Set_* \rightarrow Pf$ such that $F \circ G = I$ and $G \circ F = I$, i was thinking in find some obligatory definitions, but appear to me that don't exists.

• I wouldn't try contradiction. I'd try to find a property which is not shared by these categories. Feb 25 '15 at 21:35

In $Set_*$ every singleton (a,a) is a zero object. If both categories are isomorphic , the number of zero objects must be the same. The functor $F$ then must send every zero object to a zero object.