We say that a digraph is a $(t,λ)$-liking digraph if every $t$ vertices have exactly $λ$ common out-neighbors. In 1975, Plesník [Graphs with a homogeneity, 1975. {\it Glasnik Mathematicki} 10:9-23] proved that any $(t,1)$-liking digraph is the complete digraph on $t+1$ vertices for each $t\geq 3$. Choi {\it et al}. [A digraph version of the Friendship Theorem, 2025. {\it Discrete mathematics}, 348(1), 114238] showed that a $(2,1)$-liking digraph is a fancy wheel digraph or a $k$-diregular digraph for some positive integer $k$. In this paper, we extend these results by completely characterizing the $(t,λ)$-liking digraphs with $t \geq λ+2$ and giving some equivalent conditions for a $(t,λ)$-liking digraph being a complete digraph on $t+λ$ vertices.