For a matrix coalgebra $C$ over some field, we determine all small
subcoalgebras of the free Hopf algebra on $C$, the free Hopf algebra with a
bjective antipode on $C$, and the free Hopf algebra with antipode $S$
satisfying $S^{2d}={\rm id}$ on $C$ for some fixed $d$. We use this information
to find the endomorphisms of these free Hopf algebras, and to determine the
centers of the categories of Hopf algebras, Hopf algebras with bijective
antipode, and Hopf algebras with antipode of order dividing 2d.