  To each symmetrizable Cartan matrix, we associate a finite free EI category.
We prove that the corresponding category algebra is isomorphic to the algebra
defined in [C. Geiss, B. Leclerc, and J. Schr\"{o}er, Quivers with relations
for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017),
61--158], which is associated to another symmetrizable Cartan matrix. In
certain cases, the algebra isomorphism provides an algebraic enrichment of the
well-known correspondence between symmetrizable Cartan matrices and graphs with
automorphisms.
