Let \(LG\) denote the loop group of a compact connected Lie group \(G\), i.e., the group of smooth maps \(l : S^1\to G\) where multiplication is defined pointwise. We assume that \(\pi_1(G)\) is torsion free and \(S^1\) acts on \(LG\) via transformation such that \((\theta_0^\ast l)(\theta)=l(\theta-\theta_0)\). This \(S^1\) is written as \(\mathbb{T}\). Let \({\mathcal H}\) be a complex separable Hilbert space. Then a projective representation \(\rho : LG \to PU({\mathcal H})\) is called a positive projective representation if it satisfies the conditions (i) the action of \(\mathbb{T}\) lifts to the central extension \(\widehat{LG}:=LG\times_{PU({\mathcal H})}U({\mathcal H})\) of \(LG\) by \(U(1)\), which then allows us to define the semidirect product \(\widehat{LG}\rtimes\mathbb{T}\); (ii) for this group, there exists a lifting \(\hat{\rho} : \widehat{LG}\rtimes\mathbb{T}\to U({\mathcal H})\) of \(\rho\) such that \(\hat{\rho}(\theta_0)\) acts as an operator \(A\) with positive spectrum. Moreover, given a positive central extension \[ 1\to U(1) \to LG^\tau \to LG \to 1 \] of \(LG\) by \(U(1)\) (definition omitted), a positive projective representation \(\rho\) is said to be at level \(\tau\) if it holds that \(\widehat{LG}\cong LG^\tau\). We write \(R^\tau(LG)\) for the Grothendieck group of representations of \(LG\) at level \(\tau\) and we let the letter \(\tau\) denote the twisting arising from the positive central extension \(LG^\tau\) given above. Let \(K^{\tau+\sigma+k}_G(G)\) be the twisted equivariant \(K\)-group of \(G\) where \(G\) acts on itself by conjugation (here \(\sigma\) is a special central extension, called the spin extension). We then have an isomorphism \[ FHT_G : R^\tau(LG) \to K_G^{\tau+\sigma+\text{rank}(G)}(G) \] which is due to D. S. Freed et al. [J. Am. Math. Soc. 26, No. 3, 595–644 (2013; Zbl 1273.22015)]. The main purpose of this paper is to prove naturality of this isomorphism for tori. It can be stated more precisely as follows. Let \(f : T \to S\) be a smooth group homomorphism between two tori \(T\) and \(S\) where \(df\) is assumed to be injective. The author first constructs for this map two types of homomorphisms \[ f^\sharp : K_T^{\tau+\text{dim}(T)}(T) \to K_S^{f^*\tau+\text{dim}(S)}(S), \qquad f^! : R^\tau(LT) \to R^{f^*\tau}(LS) \] and then verifies that these homomorphisms satisfy the equality \[ f^\sharp\circ FHT_T=FHT_S\circ f^!. \] These are accomplished by making detailed observations of the construction of the isomorphism \(FHT_G\).