Structured chain complexes V

Let $f \colon C \rightarrow D$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}f \colon C \rightarrow D$ and $g \colon D \rightarrow E$$g \colon D \rightarrow E$ be chain maps and let $j \colon gf \simeq 0 \colon C \rightarrow E$$j \colon gf \simeq 0 \colon C \rightarrow E$ be a chain homotopy from~$gf$$gf$ to~$0$$0$. Then the formula

$\displaystyle \Phi_j (y,x) = g(y) + j(x)$ $$\displaystyle \Phi_j (y,x) = g(y) + j(x)$$

gives another chain map $\Phi_j \colon \mathcal{C} (f) \rightarrow E$$\Phi_j \colon \mathcal{C} (f) \rightarrow E$ making the following diagram commutative

$\displaystyle \xymatrix{C \ar[r]^{f} & D \ar[r]^{e} \ar[dr]_{g} & \mathcal{C}(f) \ar@{-->}[d]^{\Phi_j} \\ & & E }$

Find appropriate $j \colon ef \simeq 0 \colon C \rightarrow \mathcal{C} (f)$$j \colon ef \simeq 0 \colon C \rightarrow \mathcal{C} (f)$ for which we have

$\displaystyle \mathrm{ev}_l (\delta \varphi,\varphi) = \delta\varphi_0 + f \varphi_0 \colon \mathcal{C} (f)^{n+1-\ast} \rightarrow D.$ $$\displaystyle \mathrm{ev}_l (\delta \varphi,\varphi) = \delta\varphi_0 + f \varphi_0 \colon \mathcal{C} (f)^{n+1-\ast} \rightarrow D.$$

and

$\displaystyle \mathrm{ev}_r (\delta \varphi,\varphi) = \begin{pmatrix} \delta \varphi_0 \\ \varphi_0 f^\ast \end{pmatrix} \colon D^{n+1-\ast} \rightarrow \mathcal{C} (f).$ $$\displaystyle \mathrm{ev}_r (\delta \varphi,\varphi) = \begin{pmatrix} \delta \varphi_0 \\ \varphi_0 f^\ast \end{pmatrix} \colon D^{n+1-\ast} \rightarrow \mathcal{C} (f).$$