0MXP Proposition 2.15. The functor โจMโ๐ฏHom๐โโก(M,โ):โจ๐ฏโฉโ ,โจญ,โ,โจยฑ1โฉโmodfgpโคโ-โกEnd๐โโก(๐ฏ)\bigoplus_{M\in\mathcal{T}}\operatorname{Hom}^{\bullet}_{\mathcal{C}}(M,-):\langle\mathcal{T}\rangle_{\cong,\inplus,\oplus,\langle\pm 1\rangle}\to\operatorname{mod}^{\mathbb{Z}}_{\operatorname{fgp}}\operatorname{-}\operatorname{End}_{\mathcal{C}}^{\bullet}(\mathcal{T}) is an equivalence of categories.