Is a function defined as PRF with a given key a PRG? Let $G(s)=F_{0^n}(s)$ ($ 0^n $ is a fixed key given to the PRF F) for any $n∈\mathbb{N}$ and $s∈\{0,1\}^n$.

Is G, defined via PRF as above, necessarily a pseudorandom generator?

I presume that it is a PRG, since we don't really know what is the seed s. But the fact that $ F_{0^n}() $ is known, makes me doubt myself.

I don't seek for a proof, just an intuition or a hint (i do understand how to prove such claims, i just don't get the "catch" in this specific case).
Consider a PRF $F$ which on $n$-bit keys maps $n$-bit inputs to $2n$-bit outputs. Consider $F'$ where $F'_k(x) = F_k(x)$ if $k \ne 0^n$ and $0^{2n}$ otherwise.