Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 5 / c / Solution

Identities are self-adjoint. If $f⊣g$ and $f^{\prime }⊣g^{\prime }$, thenwith the second inequality written after inserting the two units in the appropriate order. Hence $f^{\prime }f⊣g{g}^{\prime }$, so left adjoints form a subcategory.

In $\mathbf{Poset}$, these are exactly monotone maps possessing right adjoints, equivalently lower adjoints; when all joins exist, they are precisely the arbitrary-join-preserving maps.

In the inclusion-ordered category of relations, a relation $R:A⇝B$ is left adjoint exactly when it is total and single-valued. It is therefore the graph of a function, and its right adjoint is $R^{\mathrm{op}}$. This is the left adjoint relation is a function criterion.