Simple transitive 2-representations for two non-fiat 2-categories of projective functors

Abstract: We show that any simple transitive $2$-representation of the $2$-ca-te-go-ry of projective endofunctors for the quiver algebra of $\Bbbk(\xymatrix{\bullet\ar[r]&\bullet})$ and for the quiver algebra of $\Bbbk(\xymatrix{\bullet\ar[r]\ar@/^/@{.}[rr]&\bullet\ar[r]&\bullet})$ is equivalent to a cell $2$-representation.