What is meant by Lee when he says "the definition of a differential was cooked up to give a coordinate independent meaning to the Jacobian matrix"

Question

Given two smooth manifolds $M,N$ and a smooth map $F:M \to N$, it seems that the entries of the Jacobian matrix for $F$ near $a \in M$ very much should depend on the choice of coordinate charts $(\varphi, U)$ near $a$ and $(\psi,W)$ near $F(a)$, so I am confused about how the matrix should be coordinate independent.