Abstract The equalizer of a set homomorphisms $S: F(a, b)\rightarrow F(\Delta)$ has rank at most two if S contains an injective map and is not finitely generated otherwise. This proves strong form Stallings’ Equalizer Conjecture for the free group two. Results are also obtained pairs $g, h:F(\Sigma)\rightarrow when images inert in, or retracts of, $F(\Delta)$.