Show that the collection of Turing-recognizable languages is closed under homomorphism
The proof is posted by Anwar Saiah on this post. I quote it here (with its formatting improved).

Given a language $L$ that is Turing recognizable and a TM $M$ that recognizes it and a homomorphism $f$, we build a NTM $M'$ that recognizes $f(L)$.
$M'$ looks like this:
On input $w$ :

Non-deterministicly feed words from $\Sigma^∗$ to $f$ until you obtain $w$.

Run $M$ on $f^{−1}(w)$, if $M$ accepts accept, otherwise reject.



This works because $M'$ is a non-deterministic Turing machine.  A NTM accepts $w$ if there is at least one branch accepting $w$, so if there is any word $x$ such that $f(x)=w$ and $M(x)$ accepts, this will find it and accept.