What\'s the dual of a binary operation?

What\'s the dual of a binary operation? References 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?