By the way, we can write the first theorem here more tersely if we use the notion of "image" from Lecture 9. If \\(f : A \to B \\) is any function, for any subset \\(S \subseteq A \\), its **[image](https://en.wikipedia.org/wiki/Image_(mathematics)#Image_of_a_subset)** under \\(f\\) is

$$f_{!}(S) = \\{b \in B: \; b = f(a) \textrm{ for some } a \in A\\} .$$

Another way to write this is

$$f_{!}(S) = \\{f(a) : \; a \in A\\} .$$

Thus, we can take our first theorem and rewrite it this way:

**Theorem.** If a monotone function \\( f: A \to B \\) between posets is a left adjoint and the join of \\( S \subseteq A \\) exists, then

$$ f (\bigvee S ) = \bigvee f_{!} (S). $$

Similarly, if a monotone function \\(g : B \to A \\) between posets is a right adjoint and the meet of \\( S \subseteq B \\) exists, then

$$ g(\bigwedge S) = \bigwedge g_{!}(S). $$

I didn't write it this way because I thought it would confuse some people, but it's very cute.

$$f_{!}(S) = \\{b \in B: \; b = f(a) \textrm{ for some } a \in A\\} .$$

Another way to write this is

$$f_{!}(S) = \\{f(a) : \; a \in A\\} .$$

Thus, we can take our first theorem and rewrite it this way:

**Theorem.** If a monotone function \\( f: A \to B \\) between posets is a left adjoint and the join of \\( S \subseteq A \\) exists, then

$$ f (\bigvee S ) = \bigvee f_{!} (S). $$

Similarly, if a monotone function \\(g : B \to A \\) between posets is a right adjoint and the meet of \\( S \subseteq B \\) exists, then

$$ g(\bigwedge S) = \bigwedge g_{!}(S). $$

I didn't write it this way because I thought it would confuse some people, but it's very cute.