Lemma 15.87.6. Let $\mathcal{D}$ be a triangulated category. Let $(K_ n)$ and $(M_ n)$ be inverse systems of objects of $\mathcal{D}$ with derived limits $K$ and $M$. Let $a : (K_ n) \to (M_ n)$ be a pro-isomorphism of pro-objects. Then $a$ can be used to produce a (non-canonical) isomorphism $K \to M$.
Proof. We obtain an arrow $K \to M$ fitting into a morphism
\[ \xymatrix{ K \ar[r] \ar[d] & \prod K_ n \ar[r] \ar[d]^{a'} & \prod K_ n \ar[d]^{a''} \\ M \ar[r] & \prod M_ n \ar[r] & \prod M_ n } \]
of defining distinguished triangles by Derived Categories, Remark 13.34.4. Thus, for every object $L$ of $\mathcal{D}$ we obtain a map of short exact sequences
\[ \xymatrix{ 0 \ar[r] & R^1\mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(L, K_ n[-1]) \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(L, K) \ar[r] \ar[d] & \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(L, K_ n) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & R^1\mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(L, M_ n[-1]) \ar[r] & \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(L, M) \ar[r] & \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(L, M_ n) \ar[r] & 0 } \]
see Lemma 15.87.5 and its proof. By Lemma 15.87.4 the left and right vertical arrows are isomorphisms1. Thus the middle arrow is an isomorphism. By the Yoneda lemma we see that the map $K \to M$ is an isomorphism. $\square$
[1] If the pro-isomorphism $a$ is given by maps $a_ n : K_{m_ n} \to M_ n$ the we obtain a pro-isomorphism on $\mathop{\mathrm{Hom}}\nolimits $ using the corresponding maps $\mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(L, K_{m_ n}) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(L, M_ n)$. Thus the construction of the arrows on $\mathop{\mathrm{lim}}\nolimits $ and $R^1\mathop{\mathrm{lim}}\nolimits $ in the proof of Lemma 15.87.4 agrees with the construction of $a'$ and $a''$ in Derived Categories, Remark 13.34.4.
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