Is this possible: In OOP, private methods in a class form a F-coalgebra and public methods in a class form an F-algebra?

Question

I recently found out that OOP classes turn out to be F-coalgebras:

https://www.semanticscholar.org/paper/Objects-and-Classes%2C-Co-Algebraically-Jacobs/c7c45abf7d99e0aef627fd5223023bf82e70dc71

The coalgebraic perspective on objects and classes in object-oriented programming is elaborated: objects consist of a (unique) identifier, a local state, and a collection of methods described as a coalgebra; classes are coalgebraic (behavioural) specifications of objects. The creation of a 'new' object of a class is described in terms of the terminal coalgebra satisfying the specification

In here, https://stackoverflow.com/questions/16015020/what-does-coalgebra-mean-in-the-context-of-programming the top answer's author describes this class

class C
  private
    x, y  : Int
    _name : String
  public
    name        : String
    position    : (Int, Int)
    setPosition : (Int, Int) → C

as a coalgebra like this:

data C = Obj { x, y  ∷ Int
             , _name ∷ String }

and the public methods like this

position ∷ C → (Int, Int)
position self = (x self, y self)

name ∷ C → String
name self = _name self

setPosition ∷ C → (Int, Int) → C
setPosition self (newX, newY) = self { x = newX, y = newY }

My question is this: Is it possible to actually define the public methods as an F-algebra?

Would this make an OOP class actually a bialgebra like described in here? https://www.researchgate.net/publication/220976988_Categorical_Programming_with_Abstract_Data_Types

The main idea is to represent an ADT by a bialgebra, that is, an algebra/coalgebra pair with a common carrier

Which would explain why so often classes are used to define abstract data types in OOP languages?