  The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the smallest $k$
for which it admits a $k$-coloring where every color class induces an acyclic
subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several
groups of authors have recently studied the containment of directed graph
minors in digraphs with given dichromatic number. In this short note we improve
several of the existing bounds and prove almost linear bounds by reducing the
problem to a recent result of Postle on Hadwiger's conjecture.
