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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?

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    $\begingroup$ Comonoids with respect to the cartesian product are not interesting. Every object carries the structure of a comonoid in a unique way! $\endgroup$
    – Zhen Lin
    Aug 23, 2013 at 22:13

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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.)

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