Let $E$ be a topos and $X$ be an object of $E$ (the case of a morphism with a non-terminal codomain can easily be reduced to the case of a morphism with terminal codomain by replacing $E$ with its slice over the codomain). In general, the dependent product functor $E/X \to E$ will not necessarily be faithful. A sufficient, but not necessary, condition for faithfulness is that $X$ is subterminal (in which case, the dependent product functor will in fact be fully faithful, because it is right adjoint to a functor that is in turn right adjoint to another fully faithful functor).
Subterminality of $X$ is not necessary, however. An example of a non-subterminal object of a topos for which the dependent product functor from the slice category is faithful is any nontrivial group $G$ considered as an object of the topos of $G$-sets.
Some questions:
- If the dependent product functor $E/X \to E$ is faithful, must $X$ have no proper retracts (or equivalently, its only idempotent endomorphism is the identity)?
- Must the dependent product functor $E/X \to E$ in fact be monadic if it is faithful? Note that faithfulness implies conservativity because a topos is balanced.
- Also, must the functor $- \times X:E \to E/X$ be essentially surjective, so that $E/X$ is not only equivalent to the Eilenberg-Moore category but also to the Kleisli category of the monad $(-)^X$?
- If the dependent product functor $E/(X \times Y) \to E/Y$ is faithful for all objects $Y$, must $X$ be subterminal?
It is easy to see that Questions 1 and 3 are true for any group $G$ considered as an object of the topos of $G$-sets.