  In this paper we prove the equivalence among (i) the weakly coupled
worldsheet string theory described by the coset sigma model
$\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}\times S^3 \times T^4$ with
$SL(2,\mathbb{R})$ WZW level $k\geq 2$, (ii) the full near horizon theory of
the NS5 branes with $k$ NS5 branes wrapping $T^4\times S^1$, $p\gg1$ F1 strings
wrapping $S^1$ and $n$ units of momentum along the $S^1$ and (iii) the single
trace $T\bar{T}$ deformation of string theory in $AdS_3\times S^3\times T^4$.
As a check we compute the spectrum of the spacetime theory by performing BRST
quantization of the coset description of the worldsheet theory and show that it
matches exactly with the one derived in the case of single trace $T\bar{T}$
deformed string theory in $AdS_3$. Secondly, we compute the two-point
correlation function of local operators of the spacetime theory using the
worldsheet coset approach and reproduce the same two-point function from the
supergravity approach.
