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A digraph $D=(V, A)$ has a good pair at a vertex $r$ if $D$ has a pair ofarc-disjoint in- and out-branchings rooted at $r$. Let $T$ be a digraph with$t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that$H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition$Q=T[H_1,\dots , H_t]$ is a digraph with vertex set $\{u_{i,j_i}\mid 1\le i\let, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup\{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$  When $T$ is arbitrary, we obtain the following result: every strong digraphcomposition $Q$ in which $n_i\ge 2$ for every $1\leq i\leq t$, has a good pairat every vertex of $Q.$ The condition of $n_i\ge 2$ in this result cannot berelaxed. When $T$ is semicomplete, we characterize semicomplete compositionswith a good pair, which generalizes the corresponding characterization byBang-Jensen and Huang (J. Graph Theory, 1995) for quasi-transitive digraphs. Asa result, we can decide in polynomial time whether a given semicompletecomposition has a good pair rooted at a given vertex.

Arc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs