computability – Given a total recursive function, can you still calculate its fixed point?

We know from Kleene's recursion theorem that if $ f $ is recursive total, there must be an integer $ n $ For who $ varphi_n = varphi_ {f (n)} $. My question is: for every $ f $ recursive total, is there a computable (total) procedure that computes such a point?

For example, given $ f $, for each $ x $ we have that $ varphi_ {g (x)} = varphi_ {f (g (x))} $ with $ g $ recursive total.