Section Adjunct_Compose_assoc.
Context {B C D E : Category}
{F : B –≻ C} {G : C –≻ B} (adj : F ⊣ G)
{F' : C –≻ D} {G' : D –≻ C} (adj' : F' ⊣ G')
{F'' : D –≻ E} {G'' : E –≻ D} (adj'' : F'' ⊣ G'').

Theorem Adjunct_Compose_assoc :
match (Functor_assoc F F' F'') in _ = Y return Adjunct Y _ with
eq_refl =>
match eq_sym (Functor_assoc G'' G' G) in _ = Y return Adjunct _ Y with
eq_refl =>
Adjunct_Compose adj (Adjunct_Compose adj' adj'')
end
end
= Adjunct_Compose (Adjunct_Compose adj adj') adj''.