  Recently, Ivan Mihajlin and Alexander Smal proved a composition theorem of a
universal relation and some function via so called xor composition, that is
there exists some function $f:\{0,1\}^n \rightarrow \{0,1\}$ such that
$\textsf{CC}(\text{U}_n \diamond \text{KW}_f) \geq 1.5n-o(n)$ where
$\textsf{CC}$ denotes the communication complexity of the problem. In this
paper, we significantly improve their result and present an asymptotically
tight and much more general composition theorem of a universal relation and
most functions, that is for most functions $f:\{0,1\}^n \rightarrow \{0,1\}$ we
have $\textsf{CC}(\text{U}_m \diamond \text{KW}_f) \geq m+ n -O(\sqrt{m})$ when
$m=\omega(\log^2 n),n =\omega(\sqrt{m})$. This is done by a direct proof of
composition theorem of a universal relation and a multiplexor in the partially
half-duplex model avoiding the xor composition. And the proof works even when
the multiplexor only contains a few functions. One crucial ingredient in our
proof involves a combinatorial problem of constructing a tree of many leaves
and every leaf contains a non-overlapping set of functions. For each leaf,
there is a set of inputs such that every function in the leaf takes the same
value, that is all functions are restricted. We show how to choose a set of
good inputs to effectively restrict these functions to force that the number of
functions in each leaf is as small as possible while maintaining the total
number of functions in all leaves. This results in a large number of leaves.
