  This paper is a subsequent paper of arXiv:2303.07672. We will continue our
research on the subject of perfect algebraic spaces that is developed in
arXiv:2303.07672. By means of algebraic Frobenius morphisms, we define the
perfection of arbitrary algebraic spaces of prime characteristic. There is a
natural perfection functors on algebraic spaces. We prove several desired
properties of the perfection functor. This extends nearly all previous results
of the perfection functor on schemes, including the recent ones developed by
Bertapelle et al. in arXiv:1611.02060. Moreover, our theory extends the
previous one developed by Xinwen Zhu in arXiv:1707.05700v1, arXiv:1407.8519.
  The perfection functor rises the notion of perfection of sites, which enables
us to restate some of Zhu's theory in arXiv:1707.05700v1. Then we show that our
theory of perfect algebraic spaces in arXiv:2303.07672 is equivalent to Zhu's
theory in arXiv:1707.05700v1 when the base scheme is a perfect field of prime
characteristic.
