Let $E \subset K$, $E \subset L$ be field extensions. We denote by $Hom(K,L)$ the set of homomorphisms from $K$ to $L$, and by $Emb_E(K,L)$ the set of homomorphisms from $K$ to $L$ which restrict to identity on $E$.
Let $E$ be the prime subfield of both $K$ and $L$. Is $Emb_E(K,L)$=$Hom(K,L)$?
Remark: The left-in-right inclusion is trivial; I am asking if every homomorphism fixes the prime subfield $E$.
Idea for proof: Any homomorphism must fix 1, hence must fix the prime subfield. Is this correct?
(I am aware there are some similar questions but typically the statements only involve $E=\mathbb{Q}$ or $K$=$L$. I am trying to generalize these into one result, using the same proof.)