Let $S$ be a subset of vertices of a digraph $D$ such that every vertex in $V(D)\setminus S$ is adjacent from at least two vertices in $S$. Then, $S$ is said to be a double dominating set (total $2$-dominating set) if every vertex in $S$ is adjacent from at least one vertex in $S$ (the subdigraph induced by $S$ has no isolated vertices). The double domination number (total $2$-domination number) of a digraph $D$ is the minimum cardinality of a double dominating set (total $2$-dominating set) in $D$. In this work, we investigate these concepts, as two extensions of double domination in graphs to digraphs, along with the concepts $2$-limited packing and total $2$-limited packing which have close relationships with the above-mentioned concepts.
