Are the inequalities reversed in the following statements?
> "It's worth noting a little spinoff of the argument I gave. If \\( f : A \to B \\) is _any_ monotone function between posets,
>
> $$ f(a \vee a') \le f(a) \vee f(a') $$
>
> when the joins here actually exist. Assuming that \\( f \\) is a left adjoint gives us the reverse inequality.
> Similarly, if \\( g : B \to A \\) is any monotone function between posets,
>
> $$ g(b \wedge g') \ge g(b) \wedge g(b') $$
>
Because \\( a \vee a' \ge a \\) and \\( a \vee a' \ge a' \\), therefore \\( f(a \vee a') \ge f(a) \\) and \\( f(a \vee a') \ge f(a') \\) .
> "It's worth noting a little spinoff of the argument I gave. If \\( f : A \to B \\) is _any_ monotone function between posets,
>
> $$ f(a \vee a') \le f(a) \vee f(a') $$
>
> when the joins here actually exist. Assuming that \\( f \\) is a left adjoint gives us the reverse inequality.
> Similarly, if \\( g : B \to A \\) is any monotone function between posets,
>
> $$ g(b \wedge g') \ge g(b) \wedge g(b') $$
>
Because \\( a \vee a' \ge a \\) and \\( a \vee a' \ge a' \\), therefore \\( f(a \vee a') \ge f(a) \\) and \\( f(a \vee a') \ge f(a') \\) .
Version 2.1.8p2
