  A $k$-uniform family $\mathcal{F}$ is called intersecting if $F\cap F'\neq
\emptyset$ for all $F,F'\in \mathcal{F}$. The shadow family $\partial
\mathcal{F}$ is the family of $(k-1)$-element sets that are contained in some
members of $\mathcal{F}$. The shadow degree (or minimum positive co-degree) of
$\mathcal{F}$ is defined as the maximum integer $r$ such that every $E\in
\partial \mathcal{F}$ is contained in at least $r$ members of $\mathcal{F}$. In
2021, Balogh, Lemons and Palmer determined the maximum size of an intersecting
$k$-uniform family with shadow degree at least $r$ for $n\geq n_0(k,r)$, where
$n_0(k,r)$ is doubly exponential in $k$ for $4\leq r\leq k$. In the present
paper, we present a short proof of this result for $n\geq 2(r+1)^rk
\frac{\binom{2k-1}{k}}{\binom{2r-1}{r}}$ and $4\leq r\leq k$.
