In fact, the Product comonad is just the dual of the Writer monad and effectively the same as the Reader monad (both discussed below)

the ∧ and ∨ operators are dual in this sense, because (¬x ∧ ¬y) and ¬(x ∨ y) are equivalent. This means that for every theorem of classical logic there is an equivalent dual theorem. De Morgan's laws are examples

Dual (mathematics), a notion of paired concepts that mirror one another