Consider the preorder \$$P = Q = \underline{3} \$$.

1) Let \$$f , g \$$ be the monotone maps shown below:

Is it the case that \$$f\$$ is left adjoint to \$$g\$$?
Check that for each \$$1 \le p, q \le 3 \$$, one has \$$f(p) \le q \text{ iff } p \le g(q) \$$.
2) Let \$$f , g \$$ be the monotone maps shown below:
Is it the case that \$$f\$$ is left adjoint to \$$g\$$?