  We say that a digraph is a $(t,\lambda)$-liking digraph if every $t$ vertices
have exactly $\lambda$ common out-neighbors. In 1975, Plesn\'{i}k [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,
2023. arXiv preprint arXiv:2305.04058] (to appear in {\it Discrete
mathematics}) 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 $(t,\lambda)$-liking digraphs
with $t \geq \lambda+2$ and giving some equivalent conditions for a
$(t,\lambda)$-liking digraph being a complete digraph on $t+\lambda$ vertices.
