So if we have
$H_0 :\theta=\theta_0$ vs $H_1 :\theta=\theta_1$
It is easy to see that this is a case of simple vs simple hypothesis (assuming that $\theta$ is the only unknown parameter of our distribution)
$H_0 :\theta<=\theta_0$ vs $H_1 :\theta>\theta_0$
Is this composite vs composite or simple vs composite?
Since it is somewhat equivalent to
$H_0 :\theta=\theta_0$ vs $H_1 :\theta>\theta_0$
Which I guess it's a simple vs composite hypothesis
And last, if we have two unkown parameters, is $H_0 :\alpha=\alpha_0 , \beta>=\beta_0$
Simple or composite?