Comonoid object

Category of comonoid objects

The category of comonoids in $𝖢$ , denoted $𝖢 𝗈 𝗆 𝗈 𝗇 𝖢$ is a category where an object is a comonoid object in $𝖢$ and a morphism is a comonoid morphism. #m/def/cat

As the cartesian completion of a symmetric monoidal category

From the dual result, it immediately follows that if $𝖢$ is a symmetric monoidal category, then $𝖢 𝗈 𝗆 𝗈 𝗇 𝖢$ is a cartesian category under the natural tensor product. #m/thm/cat

There is an evident forgetful functor $𝖢 𝗈 𝗆 𝗈 𝗇 𝖢 \to 𝖢$ , monoidal if $𝖢$ is symmetric.
$𝖬 𝗈 𝗇 𝖢$ = $𝖢 𝗈 𝗆 𝗈 𝗇 𝖢 𝐨 𝐩 𝐨 𝐩$

#state/tidy | #lang/en | #SemBr