[proofplan] We compare the two injective resolutions by applying the injective comparison theorem to the identity map of $M$ in both directions. This gives cochain maps $\alpha^\bullet: I^\bullet \to J^\bullet$ and $\beta^\bullet: J^\bullet \to I^\bullet$ over $\operatorname{id}_M$. The two composites are then comparison maps from an injective resolution to itself over $\operatorname{id}_M$, so homotopy uniqueness of comparison maps identifies each composite with the corresponding identity map up to cochain homotopy. [/proofplan] [step:Construct a comparison map from $I^\bullet$ to $J^\bullet$ over $\operatorname{id}_M$] Let \begin{align*} \operatorname{id}_M: M &\to M \\ m &\mapsto m \end{align*} be the identity homomorphism of left $R$-modules. Since $I^\bullet$ and $J^\bullet$ are injective resolutions of $M$, the injective comparison theorem applies to the map $\operatorname{id}_M$ with source resolution $I^\bullet$ and target resolution $J^\bullet$ (citing a result not yet in the wiki: Injective Comparison Theorem). Hence there exists a cochain map \begin{align*} \alpha^\bullet: I^\bullet \to J^\bullet \end{align*} such that, in degree $0$, \begin{align*} \alpha^0 \circ \varepsilon_I = \varepsilon_J \circ \operatorname{id}_M = \varepsilon_J. \end{align*} Thus $\alpha^\bullet$ is a map of augmented cochain complexes over $M$. [/step] [step:Construct a comparison map from $J^\bullet$ to $I^\bullet$ over $\operatorname{id}_M$] Apply the same injective comparison theorem to the identity homomorphism \begin{align*} \operatorname{id}_M: M \to M \end{align*} with source resolution $J^\bullet$ and target resolution $I^\bullet$. Since $J^\bullet$ and $I^\bullet$ are injective resolutions of $M$, the hypotheses of the theorem are again satisfied. Therefore there exists a cochain map \begin{align*} \beta^\bullet: J^\bullet \to I^\bullet \end{align*} such that \begin{align*} \beta^0 \circ \varepsilon_J = \varepsilon_I \circ \operatorname{id}_M = \varepsilon_I. \end{align*} Thus $\beta^\bullet$ is also a map of augmented cochain complexes over $M$. [/step] [step:Identify the composite $\beta^\bullet \circ \alpha^\bullet$ as a comparison map over $\operatorname{id}_M$] The composite \begin{align*} \beta^\bullet \circ \alpha^\bullet: I^\bullet \to I^\bullet \end{align*} is a cochain map because it is the composite of cochain maps. Its augmentation condition is computed in degree $0$: \begin{align*} (\beta^0 \circ \alpha^0) \circ \varepsilon_I = \beta^0 \circ (\alpha^0 \circ \varepsilon_I) = \beta^0 \circ \varepsilon_J = \varepsilon_I. \end{align*} Hence $\beta^\bullet \circ \alpha^\bullet$ is a comparison map from $I^\bullet$ to itself over $\operatorname{id}_M$. The identity cochain map \begin{align*} \operatorname{id}_{I^\bullet}: I^\bullet \to I^\bullet \end{align*} also satisfies \begin{align*} \operatorname{id}_{I^0} \circ \varepsilon_I = \varepsilon_I, \end{align*} so it is another comparison map from $I^\bullet$ to itself over $\operatorname{id}_M$. [/step] [step:Homotope $\beta^\bullet \circ \alpha^\bullet$ to $\operatorname{id}_{I^\bullet}$] By homotopy uniqueness for injective comparison maps, any two comparison maps between the same injective resolutions inducing the same module homomorphism on $M$ are cochain homotopic (citing a result not yet in the wiki: Homotopy Uniqueness for Injective Comparison Maps). The maps \begin{align*} \beta^\bullet \circ \alpha^\bullet,\qquad \operatorname{id}_{I^\bullet}: I^\bullet \to I^\bullet \end{align*} are both comparison maps over $\operatorname{id}_M$, as verified above. Therefore \begin{align*} \beta^\bullet \circ \alpha^\bullet \simeq \operatorname{id}_{I^\bullet}. \end{align*} [/step] [step:Homotope $\alpha^\bullet \circ \beta^\bullet$ to $\operatorname{id}_{J^\bullet}$] The composite \begin{align*} \alpha^\bullet \circ \beta^\bullet: J^\bullet \to J^\bullet \end{align*} is a cochain map, and its augmentation condition is \begin{align*} (\alpha^0 \circ \beta^0) \circ \varepsilon_J = \alpha^0 \circ (\beta^0 \circ \varepsilon_J) = \alpha^0 \circ \varepsilon_I = \varepsilon_J. \end{align*} Thus $\alpha^\bullet \circ \beta^\bullet$ is a comparison map from $J^\bullet$ to itself over $\operatorname{id}_M$. The identity map $\operatorname{id}_{J^\bullet}: J^\bullet \to J^\bullet$ is also a comparison map over $\operatorname{id}_M$, since \begin{align*} \operatorname{id}_{J^0} \circ \varepsilon_J = \varepsilon_J. \end{align*} Applying homotopy uniqueness for injective comparison maps gives \begin{align*} \alpha^\bullet \circ \beta^\bullet \simeq \operatorname{id}_{J^\bullet}. \end{align*} [/step] [step:Conclude that the augmented injective resolutions are homotopy equivalent] We have constructed cochain maps \begin{align*} \alpha^\bullet &: I^\bullet \to J^\bullet, \\ \beta^\bullet &: J^\bullet \to I^\bullet \end{align*} of augmented complexes over $M$ such that \begin{align*} \beta^\bullet \circ \alpha^\bullet \simeq \operatorname{id}_{I^\bullet}, \qquad \alpha^\bullet \circ \beta^\bullet \simeq \operatorname{id}_{J^\bullet}. \end{align*} This is precisely the definition of a homotopy equivalence of the augmented cochain complexes $I^\bullet$ and $J^\bullet$ over $M$. [/step]