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.

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.