Functional equation on integers: $f(m\cdot n)=f(m)f(n)$ and $f(m+n)=f(m)+f(n)$

Question

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(m \cdot n) = f(m) f(n)$ for $\gcd(m, n) = 1$ and $f(m + n) = f(m) + f(n)$ for $\forall m, n \in \mathbb{P}$.

I think only solution should be $f(n) = n, \ \forall n \in \mathbb{N}$... But I only managed to show $f(3) = 3$ and for other small primes...