 $$\begin{aligned}   \xymatrix{    K\mathcal{M}(\mathfrak{g},K,I_{X_{G}}) \ar[r] & K(X_{G},K)\ar[r]^{CC} & \mathscr{L}(X_G,K)\\     K\mathcal{M}(\mathfrak{l},K_{L},I_{X_{L}}) \ar[u]^{\mathscr{R}_{(\mathfrak{l},K_{L})}^{(\mathfrak{g},K)}}\ar[r] & K(X_{L},K_{L})\ar[u]^{I_{L}^{G}}\ar[r]^{CC} & \mathscr{L}(X_{L},K_L).\ar[u]^{\left(I_{L}^{G}\right)_{\ast}}       }\end{aligned}$$ 