Prompted by an inquiry of Manin on whether a coacting Hopf-type structure [Formula: see text] and an algebra [Formula: see text] that is coacted upon share algebraic properties, we study the particular case of [Formula: see text] being a path algebra [Formula: see text] of a finite quiver [Formula: see text] and [Formula: see text] being Hayashi’s face algebra [Formula: see text] attached to [Formula: see text]. This is motivated by the work of Huang, Wicks, Won and the second author, where it was established that the weak bialgebra coacting universally on [Formula: see text] (either from the left, right, or both sides compatibly) is [Formula: see text]. For our study, we define the Kronecker square [Formula: see text] of [Formula: see text], and show that [Formula: see text] as unital graded algebras. Then we obtain ring-theoretic and homological properties of [Formula: see text] in terms of graph-theoretic properties of [Formula: see text] by way of [Formula: see text].