Sampling distribution of a functional T

Question

While studying the bootstrap method, I came across with the following definition of the sampling distribution of a functional T:

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Let's say $X_1,...,X_n$ are i.i.d with distribution function $F$, then we estimate $T(F)$ by $T(F_n) \equiv T_n$, where $F_n$ is the empirical d.f.

Consider the estimation of:

$H_n(x,F) \equiv P_F(\sqrt n(T_n - T(F)) \leq x).$

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And then they explain how we can estimate that function.

My question is, why can we define the sampling distribution of a functional like that?

Thanks