Monoid object ($(A,m,j)$)

An object with multiplication $m:A\otimes A\to A$ and unit $j:I\to A$ satisfying associativity and unit diagrams using the ambient associators and unitors.

A monoid object $A$ with a morphism $\epsilon :A\to I$ such that $\epsilon m$ and a suitable coevaluation make $(A,A)$ a dual pair in a monoidal category. In a category of vector spaces this is a Frobenius algebra. This condition alone is not the usual separability condition.

An object that is both a monoid object and a comonoid in a braided monoidal category, with $\Delta ,\epsilon$ preserving multiplication and unit. The product on its tensor square uses the ambient braiding.

In a symmetric monoidal category, a convolution-invertible scalar pairing $\gamma :H\otimes H\to I$ on a bimonoid inducing the comodule braiding $u\otimes v\mapsto \sum v_{(0)}\otimes u_{(0)}\gamma (u_{(1)},v_{(1)})$. Its multiplicativity, normalization and commutation axioms express the hexagon, unit and colinearity conditions. The notation abbreviates morphisms and ambient symmetries, not a requirement for elements.

A morphism $f:A\to B$ satisfying $f{m}_A=m_B(f\otimes f)$ and $f{j}_A=j_B$.