Products of digraphs and their competition graphs

If D = (V,A) is a digraph, its competition graph (with loops) CG^l(D) has the vertex set V and {u, v} ⊆ V is an edge of CG^l(D) if and only if there is a vertex w ∈ V such that (u,w), (v,w) ∈ A. In CG^l(D), loops {v} are allowed only if v is the only predecessor of a certain vertex w ∈ V. For several products D₁ ο D² of digraphs D₁ and D₂, we investigate the relations between the competition graphs of the factors D₁, D₂ and the competition graph of their product D₁ ο D₂.