Is the category, $\textbf{FHILB}$, of finite dimensional Hilbert spaces and linear maps locally regular, where `locally regular' is defined like this 


<http://ncatlab.org/nlab/show/locally+regular+category> 



Is $\textbf{FHILB}$ locally regular?
Yes. Since it has a terminal object (namely the zero-dimensional space 0=$\{0\}$), the slice category $\mathbf{FHilb} / 0$ is isomorphic to $\mathbf{FHilb}$ itself. Because all slices of a locally regular category are regular, local regularity is equivalent to regularity.


And in fact, $\mathbf{FHilb}$ is regular. It is finitely complete and cocomplete, so definitely has coequalizers of kernel pairs. Regular epimorphisms are surjections, which are stable under pullback, because pullbacks are computed as in $\mathbf{Set}$.


