Show that if \(P \xrightarrow{f}Q\) has a right adjoint g, then it is unique up to isomorphism. Is the same true for left adjoints?
Suppose \(h\) is also right adjoint to \(f\).
What it means for \(h \cong g\):
\(\forall q \in Q: h(q) \cong g(q)\)
\(g(q) \leq h(q)\)
Substitute \(g(q)\) for \(p\) in \(p \leq h(f(p))\) (from \(h\)’s adjointness) to get \(g(q) \leq h(f(g(q)))\)
Also apply \(h\) to both sides of \(f(g(q)) \leq q\) (from \(g\)’s adjointness) to get \(h(f(g(q)))\leq h(q)\)
The result follows from transitivity.
By symmetry (nothing was specified about \(h\) or \(g\)) the proof of \(h(q)\leq g(q)\) is the same.
Same reasoning for left adjoints.