Double dual mappings
Solution 1:
What you need to show is that $\Phi^{TT}( \iota_V(v))(f)=f(\Phi(v))$. Notice that $\Phi^{TT}(\iota_V(v))=\iota_V(v)\circ \Phi^T$, which maps $f$ to $\iota_V(v)(\Phi^T(f))=\iota_V(v)(f\circ \Phi)=f(\Phi(v))$.
Double dual mappings
Solution 1:
What you need to show is that $\Phi^{TT}( \iota_V(v))(f)=f(\Phi(v))$. Notice that $\Phi^{TT}(\iota_V(v))=\iota_V(v)\circ \Phi^T$, which maps $f$ to $\iota_V(v)(\Phi^T(f))=\iota_V(v)(f\circ \Phi)=f(\Phi(v))$.