Let $F$ be the free group of rank $2$ (or any finite rank if this does not matter). The set of injective group endomorphisms $F\to F$ forms a monoid $M$ by compositions. Is there a **simple looking** set of generators of $M$?

It is natural to ask this. For example if we would like to show some property $P$ holds for all injections in $M$ and is preserved under compositions, it is natural to prove $P$ for a set of generators. So by **simple looking**, I mean it would really save us some work by only proving $P$ for generators, for example, the generators all look similar to each other (like generating the mapping class groups by Dehn twists), or fall into finitely many (infinite) classes.

Obviously, $M$ contains $\mathrm{Aut}(F)$, which has a nice finite generating set: the elementary Nielsen transformations.

Also note that, $M$ is not finitely generated, since each homomorphism $F\to F$ induces a homomorphism on the abelianization $\mathbb{Z}^2\to \mathbb{Z}^2$, whose determinant could be any integer. Thus the determinants of any generating set of $M$ should be a generating set of the monoid $\mathbb{Z}$, where the operation is multiplication.

Any answer to the related question of generating the monoid of all (not necessarily injective) endomorphisms would be helpful as well. In the rank $2$ case, the non-injective ones are easily understood.

I don't have a good idea to think about this. Are there any related reference or theory? Thanks!