A well-known example is the equivalence between groups and
and one-object categories with all morphisms invertible
(i.e. all morphisms are isomorphisms).  The group is finite
**iff** the category is.

A more exotic example of a finite category with all morphisms invertible is
<a href="http://dash.harvard.edu/handle/1/2794826">Conway's $M_{13}$ groupoid</a>
(see also <a href="https://golem.ph.utexas.edu/category/2013/02/m13.html">this
blog entry</a> by <a href="http://mathoverflow.net/users/2893/john-baez">**John Baez**</a>, 
who is fond of most things categorical).
Conway calls it "$M_{13}$" because there are $13$ objects each of whose
automorphism groups is isomorphic with the Mathieu group $M_{12}$ 
(and any two objects are connected by $|M_{12}|$ isomorphisms).