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2 Fixed typo

Let $M$ be an object in an k-linear $k$-linear abelian category with enough projectives. Then one can construct an $A_\infty$-structure on the Ext algebra $$Ext^\bullet(M,M)$$ as follows: One chooses projective resolution $P\rightarrow M$ and forms the Hom complex $$Hom^\bullet (P,P)$$ Now the cohomology of this compex complex is the Ext algebra and in the case where k $k$ is a field, one can choose a "homotopy retraction" $Ext^\bullet(M,M)\rightarrow Hom^\bullet (P,P)$ and transfer the dg-algebra structure on $Hom^\bullet (P,P)$ along it.

My questions are:

1. Why does this construction (up to $A_\infty$-isomorphism) not depend on a choice of projective resolution?
2. One could try the same thing with an injective resolution, why is the result still the same?
1

# $A_\infty$ structure on Ext-algebras well defined?

Let $M$ be an object in an k-linear abelian category with enough projectives. Then one can construct an $A_\infty$-structure on the Ext algebra $$Ext^\bullet(M,M)$$ as follows: One chooses projective resolution $P\rightarrow M$ and forms the Hom complex $$Hom^\bullet (P,P)$$ Now the cohomology of this compex is the Ext algebra and in the case where k is a field, one can choose a "homotopy retraction" $Ext^\bullet(M,M)\rightarrow Hom^\bullet (P,P)$ and transfer the dg-algebra structure on $Hom^\bullet (P,P)$ along it.

My questions are:

1. Why does this construction (up to $A_\infty$-isomorphism) not depend on a choice of projective resolution?
2. One could try the same thing with an injective resolution, why is the result still the same?