Right and Left Square matrix inverses
I would like to intuitively understand why the following is true:
Let A be a square, invertible matrix. let C be A's left inverse, and B A's right inverse.
Then it follows: B=C
I am aware of the simple proof:
yet, this gives away no intuition about the logic behind this claim.
The left inverse is acting on A's columns, the right one is acting on A's rows.
This seems almost like magic.
How would you explain this in child's terms?