Fix a field $k$ of characteristic $0$. Let $X = \operatorname{Spec} A$ be an affine derived scheme of finite type, i.e., $A$ is a cdga such that $H^0(A)$ is a finitely generated over $k$ and $H^{i}(A)$ is finitely generated over $H^0(A)$ for $i<0$. Assume $A$ is quasi-free so the cotangent complex $\mathbb{L}_A$ can be computed by the K\"{a}her differential $\Omega_A$ as in the classical case. The odd tangent bundle $\mathbb{T}X[-1]$ is defined to be the relative affine scheme $\operatorname{Spec}_{A} (\operatorname{Sym}^\bullet \Omega_A[1])$. Denote $\mathbb{A}[1]$ the derived stack defined by $\mathbb{A}[1](S) = Hom_{DGA_k}(k[\eta],S)$ where $\eta$ is of $\deg 1$.
I'm looking for a proof of the isomorphism: $$ Map(\mathbb{A}[1],X) \cong \mathbb{T}X[-1]$$ where $Map$ denotes the mapping stack. I know one can show that they have the same fiber over each $S$-point $x : \operatorname{Spec}(S) \rightarrow X$ by using the universal property of the cotangent complex. But I don't understand how one can get a global map in the first place? Reference or explanations are both welcome.