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Consider the functor \( G : \textbf{Gr} \rightarrow \textbf{DDS} \) given by sending ‘source’ to ‘next’ and sending ‘target’ to the identity on ‘State’. Migrate the same data using \(G\). Write down the tables and draw the corresponding graph.
\( I : \textbf{DDS} \rightarrow \textbf{Set} \)
$$ \begin{array}{c | c} State & next \\ \hline 1 & 4 \\ 2 & 4 \\ 3 & 5 \\ 4 & 5 \\ 5 & 5 \\ 6 & 7 \\ 7 & 6 \end{array} $$
Comments
$$ \textbf{Gr} \overset{G}{\rightarrow} \textbf{DDS} \overset{I}{\rightarrow} \textbf{Set} $$ \( G.I : \textbf{Gr} \rightarrow \textbf{Set} \) $$ \begin{matrix} \begin{array}{c | c c} \text{Arrow} & \text{source} & \text{target} \\ \hline 1 & 4 & 1 \\ 2 & 4 & 2 \\ 3 & 5 & 3 \\ 4 & 5 & 4 \\ 5 & 5 & 5 \\ 6 & 7 & 6 \\ 7 & 6 & 7 \end{array} & \begin{array}{c} \text{Vertex} \\ \hline 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \\ 7
\end{array} \end{matrix} $$ \( G.K : \textbf{Gr} \rightarrow \textbf{Drawings} \)
There is a functor that maps the set-instance functor \( GI : \textbf{Gr} \rightarrow \textbf{Set} \) to a diagram-instance functor \( GK : \textbf{Gr} \rightarrow \textbf{Drawing} \)?
\[ \textbf{Gr} \overset{G}{\rightarrow} \textbf{DDS} \overset{I}{\rightarrow} \textbf{Set} \] \\( G.I : \textbf{Gr} \rightarrow \textbf{Set} \\) \[ \begin{matrix} \begin{array}{c | c c} \text{Arrow} & \text{source} & \text{target} \\\\ \hline 1 & 4 & 1 \\\\ 2 & 4 & 2 \\\\ 3 & 5 & 3 \\\\ 4 & 5 & 4 \\\\ 5 & 5 & 5 \\\\ 6 & 7 & 6 \\\\ 7 & 6 & 7 \end{array} & \begin{array}{c} \text{Vertex} \\\\ \hline 1 \\\\ 2 \\\\ 3 \\\\ 4 \\\\ 5 \\\\ 6 \\\\ 7 \end{array} \end{matrix} \] \\( G.K : \textbf{Gr} \rightarrow \textbf{Drawings} \\)  There is a functor that maps the set-instance functor \\( GI : \textbf{Gr} \rightarrow \textbf{Set} \\) to a diagram-instance functor \\( GK : \textbf{Gr} \rightarrow \textbf{Drawing} \\)?We pulled back I along G.
We _pulled back_ I along G. 