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Awodey did mean the adjunction you described. composing with $\alpha$ can be thought of as a function sending morphisms with codomain $A$ to morphisms with codomain $C$, but can easily be upgraded to a functor $Slice_A \to Slice_C$ (by the way more common notation for these slices involves the name of the category, $\mathrm E$ in your example: $\mathrm ...

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