A question involving chiral transformations and gamma matrices

I’m looking at a calculation that involves an infinitesimal transformation of a Dirac fermion field:

$$Psi rightarrow e^{i beta gamma^5} Psi.$$

Then the conjugate field $bar{Psi} = Psi^{dagger} gamma^0$ transforms as $bar{Psi} rightarrow (e^{i beta gamma^5} Psi)^dagger gamma^0$. Then from here we get:

$$Psi^dagger e^{-i beta gamma^5} gamma^0.$$

So far I understand the steps, but I don’t how from here one jumps to
$$Psi^dagger gamma^0 e^{i beta gamma^5}.$$

Why does the sign in the exponential changes and the gamma matrix is suddenly on the right?