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).