I'm having trouble with a problem from Morandi's Field & Galois Theory: Let $K$ be an algebraically closed field, and let $F$ be a subfield of $K$ with $\text{trdeg}(K/F)=\infty$. Show that there is an $F-$homomorphism $\varphi:K \to K$ that is not an $F$-automorphism.
I know for a fact that if $\text{trdeg}(K/F)<\infty$ this is not possible because we can permute the elements of the basis and every homomorphism would me surjective.
Can anybody give me an example of such homomorphism? I tried sending every trascendental to $0$, and sending some $k$-th trascendental to a ($k+1$)th one but I always have the same problem trying to make sense of $\varphi(a+b)$ and $\varphi(ab)$ when a,b are both trascendental over $F$