Yep, President's take the credit or heat for the legislation that they sign.  They however do not write it, they can make suggestions and ultimately choose to sign or not sign legislation, but the reality is that most Presidents sign the majority of what comes across their desks.  Funny how that works, our system of checks and balances means that a President does have the final say as to whether or not legislation becomes law, that's why they get the credit, good or bad, but they do not write that legislation.  Is this hard to grasp in some way?