If B(C) is projective and B_n(C') injective for each n, the canonical homomorphism λ ( C , C ′ ) : H ( H o m g r A ( C , C ′ ) ) → H o m g r A ( H ( C ) , H ( C ′ ) ) \lambda(C, C') : \mathrm{H}(\mathrm{Homgr}_A(C, C')) \to \mathrm{Homgr}_A(\mathrm{H}(C), \mathrm{H}(C')) λ ( C , C ′ ) : H ( Homgr A ( C , C ′ )) → Homgr A ( H ( C ) , H ( C ′ ))
Indeed, by X, p. 35, remarks a) and b), there exist homologisms
φ : C → H ( C ) and φ ′ : H ( C ′ ) → C ′
\varphi : C \to \mathrm{H}(C) \quad \text{and} \quad \varphi' : \mathrm{H}(C') \to C'
φ : C → H ( C ) and φ ′ : H ( C ′ ) → C ′ such that
H ( φ ) = 1 H ( C ) \mathrm{H}(\varphi) = 1_{\mathrm{H}(C)} H ( φ ) = 1 H ( C ) and
H ( φ ′ ) = 1 H ( C ′ ) \mathrm{H}(\varphi') = 1_{\mathrm{H}(C')} H ( φ ′ ) = 1 H ( C ′ ) . In the commutative diagram
H ( H o m g r A ( H ( C ) , H ( C ′ ) ) ) → H ( H o m g r ( φ , φ ′ ) ) H ( H o m g r A ( C , C ′ ) ) λ ( H ( C ) , H ( C ′ ) ) ↓ ↓ λ ( C , C ′ ) H o m g r A ( H ( C ) , H ( C ′ ) ) → H o m g r ( H ( φ ) , H ( φ ′ ) ) H o m g r A ( H ( C ) , H ( C ′ ) ) ,
\begin{array}{ccc}
\mathrm{H}(\mathrm{Homgr}_A(\mathrm{H}(C), \mathrm{H}(C'))) & \xrightarrow{\mathrm{H}(\mathrm{Homgr}(\varphi, \varphi'))} & \mathrm{H}(\mathrm{Homgr}_A(C, C')) \\
\lambda(\mathrm{H}(C), \mathrm{H}(C')) \downarrow & & \downarrow \lambda(C, C') \\
\mathrm{Homgr}_A(\mathrm{H}(C), \mathrm{H}(C')) & \xrightarrow{\mathrm{Homgr}(\mathrm{H}(\varphi), \mathrm{H}(\varphi'))} & \mathrm{Homgr}_A(\mathrm{H}(C), \mathrm{H}(C')) ,
\end{array}
H ( Homgr A ( H ( C ) , H ( C ′ ))) λ ( H ( C ) , H ( C ′ )) ↓ Homgr A ( H ( C ) , H ( C ′ )) H ( Homgr ( φ , φ ′ )) Homgr ( H ( φ ) , H ( φ ′ )) H ( Homgr A ( C , C ′ )) ↓ λ ( C , C ′ ) Homgr A ( H ( C ) , H ( C ′ )) , λ ( H ( C ) , H ( C ′ ) ) \lambda(\mathrm{H}(C), \mathrm{H}(C')) λ ( H ( C ) , H ( C ′ )) is bijective and
H o m g r ( H ( φ ) , H ( φ ′ ) ) \mathrm{Homgr}(\mathrm{H}(\varphi), \mathrm{H}(\varphi')) Homgr ( H ( φ ) , H ( φ ′ )) is the identity, whence the assertion.
Homgr A n − 1 ( Z , H ′ ) → Homgr n − 1 ( i , 1 ) Homgr A n − 1 ( B , H ′ ) → φ n − 1 ∏ p + q = n − 1 Ext 1 ( H p , H q ) ⟶ 0 ↑ λ Z n − 1 ↑ λ B n − 1 H n − 1 ( Homgr A ( Z , C ′ ) ) → H n − 1 ( Homgr ( i , 1 ) ) H n − 1 ( Homgr A ( B , C ′ ) ) → H n ( Homgr ( δ , 1 ) ) H n ( Homgr A ( C , C ′ ) ) → H n ( Homgr ( j , 1 ) ) H n ( Homgr A ( Z , C ′ ) ) → H n ( Homgr ( i , 1 ) ) H n ( Homgr A ( B , C ′ ) ) ↑ λ C n ↑ λ Z n ↑ λ B n 0 ⟶ Homgr A n ( H , H ′ ) → Homgr n ( j , 1 ) Homgr A n ( Z , H ′ ) → Homgr n ( i , 1 ) Homgr A n ( B , H ′ )
\begin{array}{ccccccccc}
\operatorname{Homgr}_{A}^{n-1}(Z,H')&
\xrightarrow{\operatorname{Homgr}^{n-1}(i,1)}&
\operatorname{Homgr}_{A}^{n-1}(B,H')&
\xrightarrow{\varphi^{n-1}}&
\displaystyle\prod_{p+q=n-1}\operatorname{Ext}^{1}(H_p,H^q)&
\longrightarrow&0
\\[2ex]
\uparrow\scriptstyle{\lambda_Z^{n-1}}&&
\uparrow\scriptstyle{\lambda_B^{n-1}}&&&&
\\[-1ex]
H^{n-1}(\operatorname{Homgr}_{A}(Z,C'))&
\xrightarrow{H^{n-1}(\operatorname{Homgr}(i,1))}&
H^{n-1}(\operatorname{Homgr}_{A}(B,C'))&
\xrightarrow{H^n(\operatorname{Homgr}(\delta,1))}&
H^n(\operatorname{Homgr}_{A}(C,C'))&
\xrightarrow{H^n(\operatorname{Homgr}(j,1))}&
H^n(\operatorname{Homgr}_{A}(Z,C'))&
\xrightarrow{H^n(\operatorname{Homgr}(i,1))}&
H^n(\operatorname{Homgr}_{A}(B,C'))
\\[2ex]
&&&&
\uparrow\scriptstyle{\lambda_C^n}&&
\uparrow\scriptstyle{\lambda_Z^n}&&
\uparrow\scriptstyle{\lambda_B^n}
\\[-1ex]
&&&
0\longrightarrow&
\operatorname{Homgr}_{A}^{n}(H,H')&
\xrightarrow{\operatorname{Homgr}^{n}(j,1)}&
\operatorname{Homgr}_{A}^{n}(Z,H')&
\xrightarrow{\operatorname{Homgr}^{n}(i,1)}&
\operatorname{Homgr}_{A}^{n}(B,H')
\end{array}
Homgr A n − 1 ( Z , H ′ ) ↑ λ Z n − 1 H n − 1 ( Homgr A ( Z , C ′ )) Homgr n − 1 ( i , 1 ) H n − 1 ( Homgr ( i , 1 )) Homgr A n − 1 ( B , H ′ ) ↑ λ B n − 1 H n − 1 ( Homgr A ( B , C ′ )) φ n − 1 H n ( Homgr ( δ , 1 )) 0 ⟶ p + q = n − 1 ∏ Ext 1 ( H p , H q ) H n ( Homgr A ( C , C ′ )) ↑ λ C n Homgr A n ( H , H ′ ) ⟶ H n ( Homgr ( j , 1 )) Homgr n ( j , 1 ) 0 H n ( Homgr A ( Z , C ′ )) ↑ λ Z n Homgr A n ( Z , H ′ ) H n ( Homgr ( i , 1 )) Homgr n ( i , 1 ) H n ( Homgr A ( B , C ′ )) ↑ λ B n Homgr A n ( B , H ′ ) Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
Nothing.
Nothing.