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.

[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.