I have defined:
if $i: K \to \hat{K}$ and $j: L\to \hat{L}$ are two field extensions we say that they are isomorphic if there is a $\lambda: K \to L$ and a $\mu:\hat{K} \to \hat{L}$ such that $\mu(i(k)) = j(\lambda(k))$ for all $k \in K$.
I am then told that if $K \subset \hat{K}$ and $L \subset \hat{L}$ then we have that the restriction of $\mu$ to $K$ is $\lambda$ i.e. $\mu |_K = \lambda$. I really don't see how this follows, it seem it would follow if $i(k) = k$ and $j(\lambda(k)) = \lambda(k)$ but I don't see how that would be true at all
What you're told is slightly sloppily worded. The assumption $K\subset \hat K$ should really read "$K\subset \hat K$ and $i$ is the identity map", and similarly for $L$.
(This is not uncommonly left implicit, because there would be no reason to have $K$ be an actual subset of $\hat K$ if not because $i$ was also the identity).
Your definition should probably also include a requirement that $\lambda$ and $\mu$ are bijections.