A dominating (respectively, total dominating) set S of a digraph D is a set of vertices in D such that the union of the closed (respectively, open) out‐neighborhoods of vertices in S equals the vertex set of D. The minimum size of a dominating (respectively, total dominating) set of D is the domination (respectively, total domination) number of D, denoted γ(D) (respectively, γt(D)). The maximum number of pairwise disjoint closed (respectively, open) in‐neighborhoods of D is denoted by ρ(D) (respectively, ρo(D)). We prove that in digraphs whose underlying graphs have girth at least 7, the closed (respectively, open) in‐neighborhoods enjoy the Helly property, and use these two results to prove that in any ditree T (i.e., a digraph whose underlying graph is a tree), γt(T)=ρo(T) and γ(T)=ρ(T). By using the former equality we then prove that γt(G×T)=γt(G)γt(T), where G is any digraph and T is any ditree, each without a source vertex, and G×T is their direct product. From the equality γ(T)=ρ(T) we derive the bound γ(G□T)≥γ(G)γ(T), where G is an arbitrary digraph, T an arbitrary ditree and G□T is their Cartesian product. In general digraphs this Vizing‐type bound fails, yet we prove that for any digraphs G and H, where γ(G)≥γ(H), we have γ(G□H)≥12γ(G)(γ(H)+1). This inequality is sharp as demonstrated by an infinite family of examples. Ditrees T and digraphs H enjoying γ(T□H)=γ(T)γ(H) are also investigated. [ABSTRACT FROM AUTHOR]