Given any regular language L, we define $$shrink(L) = \{ \sigma_{1}\sigma_{2}\sigma_{3}...\sigma_{n} : \sigma_{1}\sigma_{1}\sigma_{2}\sigma_{2}\sigma_{3}\sigma_{3}...\sigma_{n}\sigma_{n} \in L \} $$
The question is whether these languages are regular or not.
The intuition says they are, and I was thinking about proving it with Homomorphism, i.e defining homomorphism from shrink(L) by duplicating each word and then proving that $h(shrink(L))$ is regular therefore shrink(L) is regular.
But I'm not exactly sure how to do it properly..
edit: another idea I had is by using DFA - if L is regular it has a DFA called M, so I can build M' by changing Delta such that $\delta (q,\sigma w) = \delta(\delta(q,\sigma w),\sigma w)$. Will it work?
Thanks for any help