  We study the map between two descriptions of the $T\bar{T}$ deformation of
conformal field theory (CFT): One is the defining description as a deformation
of CFT by the $T\bar{T}$-operator. The other is an alternative description as
the undeformed CFT on the dynamical $T\bar{T}$-deformed space that backreacts
to the state or operator insertions, reminiscent of the theory of gravity.
Instead of adopting the topological gravity description, we develop a more
literal CFT-based operator formalism that facilitates systematic and
straightforward computations of the $T\bar{T}$-deformation of the stress
tensor, operators, and their correlators, while rederiving known results in the
literature. Along the way, we discuss the backreaction to the
$T\bar{T}$-deformed space in response to local operators and exhibit the
hard-disk and free-space structures in the UV-cutoff and Hagedorn phases,
respectively, suggested by Cardy-Doyon and Jiang. To capitalize on the
alternative description of the $T\bar{T}$-deformed CFT, we focus on the
correlators of semi-heavy operators, i.e., the operators of large conformal
dimension $\Delta\gg\sqrt{c}$, and show an intuitive and simple way to obtain
the $T\bar{T}$-deformed correlators from those of the undeformed CFT on the
$T\bar{T}$-deformed space via dynamical coordinate transformations. This may
have implications in the holographic dual description, pointing towards a
working dictionary for a class of matter correlators in the cutoff AdS picture.
