I have a question, so please allow me to ask here. Thank you.
My question is:
Let $G$ be a union of directed graphs $G_1, \ldots , G_n$, that is, $G=G_1\cup\cdots\cup G_n$.
If $f:G\to G$ is a morphism of directed graphs such that $f|_{G_i}$ is an automorphism of $G_i$ for any $i=1,2,\dots, n$, then is $f$ an automorphism of $G$?