  We prove that if $G$ and $H$ are primitive strongly regular graphs with the
same parameters and $\varphi$ is a homomorphism from $G$ to $H$, then $\varphi$
is either an isomorphism or a coloring (homomorphism to a complete subgraph).
Therefore, the only endomorphisms of a primitive strongly regular graph are
automorphisms or colorings. This confirms and strengthens a conjecture of
Cameron and Kazanidis that all strongly regular graphs are cores or have
complete cores. The proof of the result is elementary, mainly relying on linear
algebraic techniques. In the second half of the paper we discuss implications
of the result and the idea underlying the proof. We also show that essentially
the same proof can be used to obtain a more general statement.
