Does the pullback of a continuous map commute with the induced map of a sheaf morphism in cohomology?

Question

Suppose we are given a continuous map $f: X\to Y$, sheaves (of abelian groups) $A,B$ over $Y$ and a morphism $\varphi: A\to B$. Does then the following diagram commute for all $n\geq 0$? $$ \require{AMScd} \begin{CD} H^n(X; f^*A) @<{f^*}<< H^n(Y;A)\\ @V{(f^*\varphi)_*}VV @V\varphi_*VV \\ H^n(X; f^*B) @<{f^*}<< H^n(Y;B) \end{CD}$$ I am reading Iversen's Cohomology of Sheaves and think I have understood how the maps $f^*$ are defined. What I am having problems with is relating the chain map between injective resolutions of $A$ and $B$ extending $\varphi$ with the one between the injective resolutions of $f^*A$ and $f^*B$ extending $f^*\varphi$.