Prove that the relation $R = \{(f,g)\in F \times F \mid \exists h \in P:(f = h^{-1}\circ g\circ h)\}$ is reflexive.

Question

Prove that the relation $R = \{(f,g)\in F \times F \mid \exists h \in P:(f = h^{-1}\circ g\circ h)\}$ is reflexive, where $F = \{ f \mid f : A \to A\}$ and $P = \{f\in F \mid f\text{ is one-to-one and onto} \}$. The original problem was to show that it is an equivalence relation, but the symmetry and transitivity were easy to prove. However, i am stuck proving that this is reflexive, or in other words I cant seem to find that function $h \in P$ for which for an arbitrary $f \in F,\ f = h^{-1}\circ f\circ h$.

Answer 1

Hint: The identity function is in $P$.