Lifting of connected geometric morphism from tripos to topos

Any geometric morphism of triposes f: P -> Q can be lifted to a geometric morphism of toposes [f]: C[P] -> C[Q]. It can also be shown that if f is an inclusion then [f] is also an inclusion. My question is the following: Suppose f: P -> Q is connected (i.e., the inverse image f^* is fully faithful, or equivalently, the unit \eta: Id => f_*f^* is an iso), is [f]: C[P] -> C[Q] then also connected? - Bodil Biering