I do not understand the following remark from the lecture:
Consider two field $L$ and $K$. We know that there exist unique(!) ring homomorphisms $f_K: \mathbb{Z} \rightarrow K$ and $f_L: \mathbb{Z} \rightarrow L$ and let $g: K \rightarrow L$ a field homomorphism. Then by uniqueness of $f_K$ and $f_L$ we have $g \circ f_K(a) = f_L(a)$ for all $a \in \mathbb{Z}$.
Remark: My definition of ring homomorphism from the lecture requires $f(1) = 1$, which is the reason of the uniqueness of $f_K$ and $f_L$.
I do not understand why $g \circ f_K(a) = f_L(a)$ should hold, could you please explain this to me?