Error: set_valued function called in context that cannot accept a set. What is it about?
The error message isn't very helpful:

regress=&gt; SELECT * FROM  compute_all_pair_by_craig(100);
ERROR:  a column definition list is required for functions returning "record"
LINE 1: SELECT * FROM  compute_all_pair_by_craig(100);


but if you rephrase the query to call it as a proper set-returning function you'll see the real problem:

regress=&gt; SELECT * FROM compute_all_pair_by_craig(100);
ERROR:  a column definition list is required for functions returning "record"
LINE 1: SELECT * FROM compute_all_pair_by_craig(100);


If you're using SETOF RECORD without an OUT parameter list you must specify the results in the calling statement, eg:

regress=&gt; SELECT * FROM compute_all_pair_by_craig(100) theresult(a integer, b integer);


However, it's much better to use RETURNS TABLE or OUT parameters. With the former syntax your function would be:

create or replace function compute_all_pair_by_craig(id_obj bigint)
    returns table(a integer, b integer) as $$
begin
    return query select o.id, generate_series(0,o.value) from m_obj as o;     
end;
$$ language plpgsql;


This is callable in SELECT-list context and can be used without creating a type explicitly or specifying the result structure at the call site.



As for the second half of the question, what's happening is that the 1st case specifies two separate columns in a SELECT-list, wheras the second returns a single composite. It's actually not to do with how you're returning the result, but how you're invoking the function. If we create the sample function:

CREATE OR REPLACE FUNCTION twocols() RETURNS TABLE(a integer, b integer) 
AS $$ SELECT x, x FROM generate_series(1,5) x; $$ LANGUAGE sql;


You'll see the difference in the two ways to call a set-returning function - in the SELECT list, a PostgreSQL specific non-standard extension with quirky behaviour:

regress=&gt; SELECT twocols();
 twocols 
---------
 (1,1)
 (2,2)
 (3,3)
 (4,4)
 (5,5)
(5 rows)


or as a table in the more standard way:

regress=&gt; SELECT * FROM twocols();
 a | b 
---+---
 1 | 1
 2 | 2
 3 | 3
 4 | 4
 5 | 5
(5 rows)