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Let $\mathsf{C}$ and $\mathsf{D}$ two categories and $\mathcal F,\mathcal G$ two functors $\mathsf{C}\rightarrow\mathsf{D}$. A natural isomorphism from $\mathcal F$ to $\mathcal G$ is the datum of a isomorphism $\nu_X:\mathcal F(X)\rightarrow \mathcal G (X)$ for every $X\in Obj(\mathsf{C})$ such that for every $\alpha\in \operatorname{Hom}(X,Y)$ in $\mathsf{C}$ we have that

$$\mathcal G(\alpha)\circ\nu_X=\nu_Y\circ\mathcal F(\alpha)$$

Now, many books say that a linear isomorphism $f$ between vector spaces is a natural isomorphism if "$f$ doesn't depend from the choice of the basis". I have two questions:

1) What does formally mean the phrase "$f$ doesn't depend from the choice of the basis"?

2) How can I match the two definitions of natural isomorphism?

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