Is a PRF always collision resistant? Context:
We usually assume that the hash functions we use in practice are both: collision resistant and pseudorandom. I wonder what's the relation between those properties.

Question:
Is a pseudo random function always collision resistant?
First, a PRF is a keyed function, meanwhile a hash function is usually keyless.  So, a hash function cannot be a PRF.

Second, a secure PRF can be swapped with a uniform random function without detection, and the best way to find a collision of a uniform random function is the birthday attack.  So, if the PRF has superpolynomially large codomain, then it would be collision-resistant, since the birthday attack would take more than polynomial-time.