Anindya - yes, for \$$f : A \to B\$$ to _have_ a right adjoint, and thus _be_ a left adjoint, it's enough for all the sets

$\\{a \in A : \; f(a) \le b \\}$

to have joins. (The "join" of any subset of \$$A\$$ is its sup, or least upper bound.) I mention this in [Lecture 17](https://forum.azimuthproject.org/discussion/2037/lecture-17-chapter-1-the-grand-synthesis#Head).

There must be some really nice relationships between this fact and topology you mention. Someone must understand them, but I don't. Does anyone here?