Comonoid ( $(C, Δ, ε)$ )

The categorical dual of a monoid object: an object with coassociative comultiplication and a counit, with the ambient constraints included.

The structure map

ε : C \to I

of a comonoid, with

(ε \otimes 1) Δ

and

(1 \otimes ε) Δ

equal to the inverse unitors. This is distinct from the counit of an adjunction.

Comultiplication ( $Δ$ )

The structure map

Δ : C \to C \otimes C

of a comonoid; it is coassociative up to the ambient associator.

f : C \to D

satisfying

Δ_{D} f = (f \otimes f) Δ_{C}

and

ε_{D} f = ε_{C}

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