Jonathan: it's good you asked about this! It's incredibly important to understand the sense in which \$$\to\$$ is adjoint to \$$\vee\$$, since it shows that _logic is built from adjoints!_

As Matthew tersely indicated, if a poset \$$A\$$ has binary meets, for every element \$$b \in A\$$ there's a monotone function

$$\cdot \wedge b : A \to A$$

sending each element \$$a \in A\$$ to \$$a \wedge b\$$. Sometimes all these monotone functions have right adjoints, which we denote by

$$b \to \cdot : A \to A$$

These send each \$$c \in A\$$ to \$$b \to c \$$. As I explained in comment #13, this occurs iff

$$a \wedge b \le c \textrm{ if and only if } a \le (b \to c)$$

for all \$$a,b,c \in A\$$. Or in words, "\$$a \wedge b\$$ implies \$$c\$$" if and only if "\$$a\$$ implies \$$b \to c \$$".

_This is the sense in which the logical connective \$$\to\$$ is the right adjoint to "and"_.

Here we have to carefully distinguish implication as a binary relation on \$$A\$$, which is \$$\le\$$, from the [material conditional](https://en.wikipedia.org/wiki/Material_conditional), a binary operation on \$$A\$$, which I'm calling \$$\to\$$ but is also called \$$\supset \$$ or other things.

\$$a \le b\$$ is a statement "external" to the poset, while \$$a \to b\$$ is "internal" to the poset: it's an element of the poset. A lot of category theory and logic involves switching back and forth between external and internal viewpoints.