  We present a new set of axioms for 2D TQFT formulated on the category of cell
graphs with edge-contraction operations as morphisms. We construct a functor
from this category to the endofunctor category consisting of Frobenius
algebras. Edge-contraction operations correspond to natural transformations of
endofunctors, which are compatible with the Frobenius algebra structure. Given
a Frobenius algebra A, every cell graph determines an element of the symmetric
tensor algebra defined over the dual space A*. We show that the
edge-contraction axioms make this assignment depending only on the topological
type of the cell graph, but not on the graph itself. Thus the functor generates
the TQFT corresponding to A.
