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I am interested in the following situation:

Suppose that $i:C \hookrightarrow to D$ is a full and faithul functor, $D$ 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 $i$ is 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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On locally-reflective subcategories

I am interested in the following situation:

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

$$C/c \to D/i(c)$$ 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 $i$ is full and faithful, but it is in my example.

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!