  We construct functors between the category of vertex algebras and that of
Costello-Gwilliam factorization algebras on the complex plane $\mathbb{C}$,
without analytic structures such as differentiable vector spaces, nuclear
spaces, and bornological vector spaces. We prove that this pair of functors is
an adjoint pair and that the functor from vertex algebras to factorization
algebras is fully faithful. Also, we identify the class of factorization
algebras that are categorically equivalent to vertex algebras. To illustrate,
we check the compatibility with the commutative structures and the
factorization algebras constructed as factorization envelopes, including the
Kac-Moody factorization algebra, the quantum observables of the $\beta\gamma$
system, and the Virasoro factorization algebra.
