copy postgresql db rows from one table to another depending on an id

I want to restructure my database. Therefore I want to move some columns from one table to another. The problem is that I do not want to simply insert the new column somehow into the destinationtable. Both tables have an id and I want to copy the column data depending on the id of both tables.

Table1:

id bigint not null
a character varying
b date
...

Table2:

id bigint not null
c character varying
d date
...

I have tried with insert into to no avail because this would add new rows in the destination table which is not what I want. Same with select into.

insert into table2 as t2 (c,d)
select t1.a, t1.b
from table1 as t1
where t1.id = t2.id