Weak four lemma with the exactness hypotheses named

Write the top row as $W\stackrel{i}{\to }X\stackrel{j}{\to }Y\stackrel{p}{\to }Z$ and the bottom row as $W^{\mathrm{\prime }}\stackrel{i^{\mathrm{\prime }}}{\to }{X}^{\mathrm{\prime }}\stackrel{j^{\mathrm{\prime }}}{\to }{Y}^{\mathrm{\prime }}\stackrel{p^{\mathrm{\prime }}}{\to }{Z}^{\mathrm{\prime }}$. Assume that exactness holds at $X$, $Y$, $X^{\mathrm{\prime }}$, and $Y^{\mathrm{\prime }}$, and that $\alpha$ and $\gamma$ are epic while $\delta$ is monic. Let $x^{\mathrm{\prime }}$ be a member of $X^{\mathrm{\prime }}$. By epicity of $\gamma$ and [L2], choose a member $y$ of $Y$ with $\gamma y\equiv j^{\mathrm{\prime }}{x}^{\mathrm{\prime }}$. Then $\delta py=p^{\mathrm{\prime }}\gamma y\equiv p^{\mathrm{\prime }}{j}^{\mathrm{\prime }}{x}^{\mathrm{\prime }}\equiv 0,$ so monicity of $\delta$ and [L1] give $py\equiv 0$. Exactness at $Y$ gives a member $x_0$ of $X$ with $j{x}_0\equiv y$ by [L3].

Assume instead that exactness holds at $X$, $Y$, $X^{\mathrm{\prime }}$, and $Y^{\mathrm{\prime }}$, that $\beta$ and $\delta$ are monic, and that $\alpha$ is epic. Let $y$ and $y^{\mathrm{\prime }}$ be members of $Y$ with $\gamma y\equiv \gamma y^{\mathrm{\prime }}$. Then $\delta py=p^{\mathrm{\prime }}\gamma y\equiv p^{\mathrm{\prime }}\gamma y^{\mathrm{\prime }}=\delta p{y}^{\mathrm{\prime }},$ so [L1] gives $py\equiv p{y}^{\mathrm{\prime }}$. By [L4], replace $y$ and $y^{\mathrm{\prime }}$ by representatives on one common epic refinement of the witnesses for both equalities and define $t:=y-y^{\mathrm{\prime }}$. Then $\gamma t=pt=0$ and $y=y^{\mathrm{\prime }}+t$ on that domain. Exactness at $Y$ gives a member $x$ of $X$ with $jx\equiv t$ by [L3].

Now $j^{\mathrm{\prime }}\beta x_0=\gamma j{x}_0\equiv \gamma y\equiv j^{\mathrm{\prime }}{x}^{\mathrm{\prime }}\mathrm{.}$ By [L4], replace $\beta x_0$ and $x^{\mathrm{\prime }}$ by representatives on a common epic domain and define $z^{\mathrm{\prime }}:=x^{\mathrm{\prime }}-\beta x_0$. Then $j^{\mathrm{\prime }}{z}^{\mathrm{\prime }}=0$ and $x^{\mathrm{\prime }}=\beta x_0+z^{\mathrm{\prime }}$ on that domain. Exactness at $X^{\mathrm{\prime }}$ gives a member $w^{\mathrm{\prime }}$ of $W^{\mathrm{\prime }}$ with $i^{\mathrm{\prime }}{w}^{\mathrm{\prime }}\equiv z^{\mathrm{\prime }}$ by [L3], and epicity of $\alpha$ gives a member $w$ of $W$ with $\alpha w\equiv w^{\mathrm{\prime }}$ by [L2]. Therefore $\beta (x_0+iw)\equiv \beta x_0+i^{\mathrm{\prime }}\alpha w\equiv \beta x_0+i^{\mathrm{\prime }}{w}^{\mathrm{\prime }}\equiv \beta x_0+z^{\mathrm{\prime }}\equiv x^{\mathrm{\prime }}\mathrm{.}$ So [L2] makes $\beta$ epic.

From step 1.2 we get $j^{\mathrm{\prime }}\beta x=\gamma jx\equiv \gamma t\equiv 0.$ Exactness at $X^{\mathrm{\prime }}$ gives a member $w^{\mathrm{\prime }}$ of $W^{\mathrm{\prime }}$ with $i^{\mathrm{\prime }}{w}^{\mathrm{\prime }}\equiv \beta x$ by [L3]. Since $\alpha$ is epic, choose $w$ in $W$ with $\alpha w\equiv w^{\mathrm{\prime }}$ by [L2]. Then $\beta (x-iw)\equiv \beta x-i^{\mathrm{\prime }}\alpha w\equiv \beta x-i^{\mathrm{\prime }}{w}^{\mathrm{\prime }}\equiv 0.$ Because $\beta$ is monic, [L1] yields $x\equiv iw$, and therefore $t\equiv jx\equiv jiw\equiv 0.$ So $y\equiv y^{\mathrm{\prime }}$, and [L1] makes $\gamma$ monic.

Hence the weak four lemma follows from the named middle-node exactness hypotheses alone.