  Let $X=(X_i)_{i\ge 1}$ and $Y=(Y_i)_{i\ge 1}$ be two sequences of independent
and identically distributed (iid) random variables taking their values,
uniformly, in a common totally ordered finite alphabet. Let LCI$_n$ be the
length of the longest common and (weakly) increasing subsequence of $X_1\cdots
X_n$ and $Y_1\cdots Y_n$. As $n$ grows without bound, and when properly
centered and normalized, LCI$_n$ is shown to converge, in distribution, towards
a Brownian functional that we identify.
