A question about the definition of PRFs Let $\mathcal{F} = \{ f_{s} \}_{s \in \{\, 0,1 \,\}^{*}}$ be a family of computable functions, where $f_{s} \colon \{\, 0,1 \,\}^{|s|} \rightarrow \{\, 0,1 \,\}^{|s|}$.

We define a family of functions $\mathcal{H} = \{ h_{s} \}_{s \in \{\, 0,1 \,\}^{*}}$, where $h_{s} \colon \{\, 0,1 \,\}^{|s|} \rightarrow \{\, 0,1 \,\}^{|s|}$ and $h_{s}\left( \cdot \right) = f_{s}\left( f_{s} \left( \cdot \right) \right)$.

Let $F_{n}$ ($H_{n}$) be the distribution of functions $f_{s}$ ($h_{s}$), where $s$ is uniformly distributed over $\{\, 0,1 \,\}^{n}$. And let $R_{n}$ denote the uniform distribution over the set of all functions from $\{\, 0,1 \,\}^{n}$ to $\{\, 0,1 \,\}^{n}$. 

We know that for any PPT adversary $A$, if for all $n$,
$$\left\vert \Pr\left[ A^{F_{n}}\left( 1^n \right) = 1 \right] - \Pr\left[ A^{R_{n}}\left( 1^n \right) = 1 \right] \right\vert \leq \epsilon (n)$$
then $\mathcal{F}$ is a family of PRFs. And if $\mathcal{F}$ is a family of PRFs, then $\mathcal{H}$ is also a family of PRFs.

How about $A$ can only can query the oracle of a function sampled only one time? In other words, if 
$$\left\vert \Pr\left[ A\left( F_{n} \right) = 1 \right] - \Pr\left[ A\left( R_{n} \right) = 1 \right] \right\vert \leq \epsilon (n)$$
holds true, can we get that
$$\left\vert \Pr\left[ A^{F_{n}}\left( 1^n \right) = 1 \right] - \Pr\left[ A^{R_{n}}\left( 1^n \right) = 1 \right] \right\vert \leq \epsilon (n)$$

PS: The title may not clear enough, but I don't know how to make it better :(. Thanks for anyone helping me to make it better :).
Let $F_n$ be a random involution induced by a random perfect matching on $\{0,1\}^n$. For any input $x$, the distributions of $F_n(x)$ and $R_n(x)$ are almost identical (in fact, $F_n(x)$ has the same distribution as $R_n(x)$ conditioned on the event $R_n(x) \neq x$, which occurs with probability $1-2^{-n}$), so they are statistically indistinguishable. However, $H_n$ is the identity function, so it is easy to distinguish from a random function.

With some more work, it might be possible to extend this from one query to any fixed polynomial number of queries.