Seeking a generalization of adjunctions

Question

We say functor $\mathbb{C} \overset{R}{\rightarrow} \mathbb{D}$ is the right adjoint to functor $\mathbb{C} \overset{L}{\leftarrow} \mathbb{D}$ when there is isomorphism between $\text{Hom} \left( L(D), C \right)$ and $\text{Hom} \left( D, R(C) \right)$ that is natural in $C$ and $D.$ My question is, is there a generalization of this concept, when the mapping between homsets is not necessarily an isomorphism ? The situation I seem to be encountering is that I have a surjection from $\text{Hom} \left( D, R(C) \right)$ to $\text{Hom} \left( L(D), C \right),$ which has a section (it could be that the surjection and its section are natural). (Also in my case $\mathbb{C} = \mathbb{D}$). Anyhow, I was just wondering if anybody knew of generalizations of the adjoint functor concept which could help me. If so, I would appreciate pointers to corresponding literature too.

Answer 1

These are known as weak adjoint functors, and were introduced in the appropriately-named 1971 paper Weak adjoint functors by Kainen. One says that $R \colon \mathbb C \to \mathbb D$ is a weak right adjoint to $L \colon \mathbb D \to \mathbb C$ if there is a surjection $\mathbb D(D, RC) \to \mathbb C(LD, C)$ natural in $C \in \mathbb C$ and $D \in \mathbb D$.