  For any $n$, we describe all endomorphisms of the braid group $B_n$ and of
its commutator subgroup $B'_n$, as well as all homomorphisms $B'_n\to B_n$.
These results are new only for small $n$ because endomorphisms of $B_n$ are
already described by Castel for $n\ge 6$, and homomorphisms $B'_n\to B_n$ and
endomorphisms of $B'_n$ are already described by Kordek and Margalit for $n\ge
7$. We use very different approaches for $n=4$ and for $n\ge 5$.
