React update function from child using context - reactjs

Question

How can i update a function from a child using context??
The context
import React from 'react';
const Fin_states = React.createContext({
title: "title_title"
});
export default Fin_states;
This is the main page , the aim is update function_() using set_function_(see the states const [function_,set_function_] ) , but doing it using the context in the List_of_list child.
import React , { useState ,useContext} from 'react';
import Input_list_panel from './Input_list_panel'
import Fin_states from './Financial_instrument_states'
export default function FinancialInstruments() {
//Attribs
const context = useContext(Fin_states);
const [value, set_value] = useState({title:"initial_title"})
const [function_, set_function_] = useState(() => (data) => console.log(data))
// set_function_ is going to be represented by update_function in the context provider.
return (
<>
<Fin_states.Provider value={{ data:value, update_value:set_value,update_function:set_function_}}>
<Input_list_panel/>
</Fin_states.Provider>
<button onClick={()=>function_("data")}>
Add item in child from parent
</button>
</>
);
}
This is the file where the function is going to be modified using useEffect in this line context.update_function(()=>updated_function)
import React , { useState , useEffect, forwardRef ,useRef,useImperativeHandle, useContext } from 'react';
import Fin_states from './Financial_instrument_states'
export default function List_of_list(props) {
const [list_of_list,set_list_of_list]= useState(["data_1","data_2"]);
const context = useContext(Fin_states);
const updated_function=(data)=>{
var new_item=[...list_of_list];
new_item.push(data)
set_list_of_list(new_item);
console.log(list_of_list)
}
useEffect(() => {
/*update_function represents at set_function_ to modify function_() state , see the context
provider in the parent FinancialInstruments*/
context.update_function(()=>updated_function)
}, [])
return (
<>
<Fin_states.Consumer>
{
(context)=>{
return(
<>
<button onClick={
()=>{
updated_function("data")
}
}>
Add item in child
</button>
</>
)
}
}
</Fin_states.Consumer>
</>
);
}
The problem is that when the state of function_() is updated in the main page (FinancialInstruments functional component) it does not use the same context than updated_function() in List_of_list.
How can i have the same result that i have in List_of_list using updated_function() but using function_() in FinancialInstruments?