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I have a functor and a prefunctor (not a functor) "in the inverse direction".

Can the notion of adjunction be generalized for prefunctors?

I remind that a prefunctor is a functor without the requirement to preserve identities (that is a prefunctor is required to preserve composition but not identities).

@Egbert asked me to describe my situation in more details. So I do below.

My real setting is the following: $(\mathsf{FCD})$ is a functor from the category $\mathsf{RLD}$ to the category $\mathsf{FCD}$ (see this preprint for definition of these categories and functions); $(\mathsf{RLD})_{\mathrm{in}}$ is a prefunctor (not a functor) from the category $\mathsf{FCD}$ to the category $\mathsf{RLD}$. I suspect that $(\mathsf{FCD})$ and $(\mathsf{RLD})_{\mathrm{in}}$ may be in some sense adjoint (but I've seen a definition of adjoints only for functors not for prefunctors).

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You can define an adjunction in any bicategory. For example, there are adjunctions in the bicategory of categories and profunctors, in the bicategory of spans of categories, etc. –  Zhen Lin Jun 5 '12 at 18:06
When this was homework we would ask questions like: what is the setting? What do you exactly mean by prefunctors? What have you tried and where did you get stuck? I think these are also relevant questions here. For the best possible answer, please try to be a bit more descriptive. –  Egbert Jun 5 '12 at 19:11
@Egbert: I've edited the question. –  porton Jun 5 '12 at 19:25

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