Category of pointed sets and category of sets are not equivalent Let $\mathbf{Set}$ denote the category of sets, with $\mathrm{hom}(X,Y)=Y^X$, and let $\mathbf{pSet}$ denote the category of pointed sets, with objects of the form $(X,x),\, x\in X$, and $\mathrm{hom}((X,x),(Y,y))=\{ f:X\to Y\mid f(x)=y\}$, then I wish to prove that there cannot exist any equivalence of these two categories.
I know how to construct equivalences, but I'm struggling to find any measure by which I can show two categories to be non-equivalent. My initial approach was to suppose the existence of an equivalence $F:\mathbf{pSet}\to\mathbf{Set}$ and reach a contradiction, but I couldn't get very far. I'm very new to category theory, so I'm not really sure how to go from here.
 A: Try looking at initial and terminal objects in both categories.
A: For variety...
In $\mathbf{Set}$, $|\hom(X,X)| = |X|^{|X|}$.
In $\mathbf{Set_*}$, $|\hom((X,x), (X,x))| = |X|^{|X \setminus x|}$
In $\mathbf{Set}$, every finite endomorphism monoid has cardinality $n^n$ for some natural number $n$, whereas in $\mathbf{Set_*}$ they all have cardinality $(n+1)^n$.
In particular, there is a set with exactly 4 endomorphisms, but no pointed set has exactly four endomorphisms. Thus there is no full and faithful functor $\mathbf{Set} \to \mathbf{Set_*}$.
A: Generally, you show that things aren't equivalent by showing that some invariant takes different values for each. In category theory these invariants typically take the form of categorical properties (properties invariant up to equivalence), such as the behavior of limits and colimits. 
Here sets and pointed sets can be distinguished by the behavior of their initial and terminal objects: the category of pointed sets has a zero object, meaning an object which is both initial and terminal, while the category of sets doesn't. 
