Is common symbol $\vdash$ an abuse of notation or there is a deep sacred connection between $$\Gamma \vdash \lambda(a:A).a:\Pi(a:A).A$$ which is preorder and $$G \vdash F \quad \mbox{($F$ is left adjoint to $G$)}$$ ?
(the source of misunderstanding is adjoined functor wiki)
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I found some useful here in Categorical Logic and Type Theory B.Jacobs (1999).
But I didn't understand about which categories author wrote.