Let [J]: Jech "Set theory" (Millenium edition)

Let $\kappa$ a limit ordinal.

From [J], T.3.11, p. 33 we have that $\kappa<\kappa^{cf(\kappa)}$.

I improved that proof, and obtain :

$\kappa < cf(\kappa)^{cf(\kappa)}$

PROOF: Let $\gamma:=cf(\kappa)$ and $c: \gamma\to \kappa$ cofinal, ordered, injective. Give a family $F=${$f_k|k<\kappa$} of functions $f_k: cf(\kappa)\to cf(\kappa)$ define $f: cf(\kappa)\to cf(\kappa)$ as follow:

$f(\delta)$ is the minimum $\beta$ such that $\beta\neq f_k(\delta)\ \forall k < c(\delta)$.

We have that $f(\delta\dot{+}1)\neq f_{c(\delta)}(\delta\dot{+}1)$ (where $\dot{+}$ is the ordinal sum, or successor).

Then $f\not\in F$. (end of Proof)

From [J], Cor.5.12 we have that $cf(2^{\aleph_\alpha})>\aleph_\alpha$ for any ordinal $\alpha$.

Form the cardinal identity $\lambda^\lambda=2^\lambda$ ([J], L.5.6) and from above follow that $2^{cf(\aleph_\alpha)}>\aleph_\alpha$

My question is:

How are related $2^{cf(\aleph_\alpha)}$ and $cf(2^{\aleph_\alpha})$ ?