# Hopf Parametric Adjoint Objects through a 2-adjunction of the type Adj-Mnd

## TL;DR

This paper explores Hopf parametric adjunctions within a specific 2-adjunction framework, revising foundational concepts and applying results to the categorical understanding of Hopf Monads.

## Contribution

It introduces a new analysis of Hopf parametric adjunctions using 2-adjunctions between adjunctions and monads, refining the theory of Hopf Monads.

## Key findings

- Revised definitions of adjoint objects in 2-categories
- Characterized adjoint objects in the 2-category of adjunctions and monads
- Applied results to categorical characterization of Hopf Monads

## Abstract

In this article Hopf parametric adjunctions are defined and analysed within the context of the 2-adjunction of the type $\mathbf{Adj}$-$\mathbf{Mnd}$. In order to do so, the definition of adjoint objects in the 2-category of adjunctions and in the 2-category of monads for $Cat$ are revised and characterized. This article finalises with the application of the obtained results on current categorical characterization of Hopf Monads.