$L=\{a^ib^i|i\geq0\}$, cfg for $L^2$ $L=\{a^ib^i|i\geq0\}$, cfg for $L^2$

can you write cfg for $L^2$ where $L=\{a^ib^i|i\geq0\}$?

the professor's answer sheet says it's $S\to AA\\ A\to aAb|\lambda$

but I think it is wrong because two $L$ have to be identical with each other.

can you help me figure out if there is cfg describing $L^2$?
The square language $L^2$ is defined as follows:
$$
L^2 = LL = \{ xy : x,y \in L \}.
$$
There is no requirement that $x = y$. Indeed, in your case $L^2 \supsetneq \{ w^2 : w \in L \}$.