The sequence given by \$$a_0=1\$$, \$$a_1=0\$$, \$$a_{n+2}=4a_n\$$, i.e., \$$1,0,4,0,16,...,0,2^{n},0,2^{n+2},...\$$ can be obtained from either the graph with two nodes \$$x\$$ and \$$y\$$ with non-identity arrows \$$g,g':x\to y\$$, \$$h,h':y\to x\$$, or from the graph with five nodes \$$x\$$, \$$y_i\$$ where \$$i\in\\{1,2,3,4\\}\$$ and arrows \$$g_i:x\to y_i\$$ and \$$h_i:y_i\to x\$$.

[Fredrick wrote](https://forum.azimuthproject.org/discussion/comment/18804/#Comment_18804):
>$J : \textbf{Graph} \rightarrow ( \mathbb{N} \rightarrow \mathbb{N} )$
Does \$$J\$$ have an inverse? Does \$$J\$$ have a right adjoint?

Since there are cases where at least two graphs give the same sequence, J is not injective, so it cannot have an inverse.