A direct approach to co-universal algebras associated to directed graphs

We prove directly that if $E$ is a directed graph in which every cycle has an entrance, then there exists a $C^*$-algebra which is co-universal for Toeplitz-Cuntz-Krieger $E$-families. In particular, our proof does not invoke ideal-structure theory for graph algebras, nor does it involve use of the gauge action or its fixed point algebra.