Right inverse is also left inverse for nonsquare matrices?

If $m≠n$ and we have the matrices $A$ $(m\times{n})$, $B$ $(n\times{m})$ and $C$ $(n\times{m})$ such that $AB=I(m\times{m})$ and $CA=I(n\times{n})$, does $B=C$?

I know the proof that it is true if we are talking about square matrices, but it doesn't help in this case.