What's the dual of a binary operation?

I have a binary operation:
$
\diamond : M\times M \to M
$
.
I want to dualize the binary operation by flipping the arrow, giving me:
$$
f : M \to M\times M
$$
Now, I can define a coassociativity law as:
$$
((f \circ fst \circ f)(m), (snd \circ f)(m))
=
((fst \circ f)(m), (f \circ snd \circ f)(m))
$$

where $fst$ extracts the first element of the tuple, and $snd$ the second. Intuitively, this coassociativity implies that the only thing you need to know about $f$ is the number of times it’s been applied.

I assume these dual constructions have been studied before, although I suspect they’ve not been called duals. What is this called, and where can I find more info? Also, this seems very different from the standard construction of comonoids to me, but could it actually be the same thing?

Solutions Collecting From Web of "What's the dual of a binary operation?"

Dualizing a binary operation $c \times c \to c$, where $c$ is an object in some category $C$, gets you a map $c \to c \sqcup c$, where $c$ is now being regarded as an object in the opposite category $C^{op}$; in particular, the product in $C$ dualizes to the coproduct in $C^{op}$. More generally, you can define comonoids with respect to any monoidal structure on a category, and then the statement is that dualizing a monoid in $C$ with respect to the cartesian monoidal structure (product) gets you a comonoid in $C^{op}$ with respect to the cocartesian (?) monoidal structure.

Coalgebras in the usual algebraic sense are comonoids with respect to the tensor product on, say, $\text{Vect}$.

As Zhen Lin mentions in the comments, a funny thing happens with the cartesian monoidal structure: every object in a category with finite products is a comonoid with respect to the cartesian monoidal structure in a unique way! The unique comultiplication is given by the diagonal map $\Delta : c \to c \times c$; see this blog post for details. (Dually, every object in a category with finite coproducts is a monoid with respect to the cocartesian monoidal structure in a unique way.)