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I am interested in the following situation:

Suppose that $i:C \to D$ is a functor, $C$ does not necessarily have a terminal object, and for each object $c$ of $C,$ the induced functor

$$C/c \to D/i(c)$$ is full and faithful and has a left-adjoint. Probably, one should say that $C$ is locally-reflective in $D$.

Has this situation been studied? It is not necessarily important that the induced functors are full and faithful, but it is in my example (but $i$ itself is not).

In such a situation, what can we say about induced functors between the presheaf categories of $C$ and $D$?

Of course, I can work this all out for myself, but, if this has already been studied, I would like to know. Thanks!

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I would probably say "locally p.r.a" (omitting the f+f condition) rather than "locally reflective", since in the case when c is terminal in C, the condition reduces not to reflectivity but to being p.r.a.: –  Mike Shulman Jun 10 '11 at 3:21
Good question. I would like to mention that if we have the stronger condition that for every diagram $c : I \to C$, the induced functor $C/c \to D/ic$ has a right adjoint, then we can conclude: If $D$ is complete, then also $C$ is complete. This can be used to show that the category of locally ringed spaces is complete, using the forgetful functor to ringed spaces ( perhaps also Hakim's thesis). –  Martin Brandenburg Jun 10 '11 at 6:28

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