Why is there no functor $\mathsf{Group}\to\mathsf{AbGroup}$ sending groups to their centers? The category $\mathbf{Set}$ contains as its objects all small sets and arrows all functions between them.  A set is "small" if it belongs to a larger set $U$, the universe.  
Let $\mathbf{Grp}$ be the category of small groups and morphisms between them, and $\mathbf{Abs}$ be the category of small abelian groups and its morphisms.  
I don't see what it means to say there is no functor $f: \mathbf{Grp} \to \mathbf{Abs}$ that sends each group to its center, when $U$ isn't even specified.  Can anybody explain?

This is very similar to Arturo Magidin's answer, but offers another point of view.
Consider the dihedral group $D_n=\mathbb Z_n\rtimes \mathbb Z_2$ with $2\nmid n$ (so the $Z(D_n)=1$). From the splitting lemma we get a short exact sequence
$$1\to\mathbb Z_n\rightarrow D_n\xrightarrow{\pi} \mathbb Z_2\to 1$$
and an arrow $\iota\colon \mathbb Z_2\to D_n$ such that $\pi\circ \iota=1_{\mathbb Z_2}$.
Hence the composite morphism
$$\mathbb Z_2\xrightarrow{\iota} D_n\xrightarrow{\pi}\mathbb Z_2$$
is an iso and would be mapped by the centre to an iso
$$\mathbb Z_2\to 1\to \mathbb Z_2$$
what is impossible. (One can also recognize a split mono and split epi above and analyze how they behave under an arbitrary functor).
Therefore the centre can't be functorial.
