Degenerate exactness criteria

Exactness of $0\to K\stackrel{u}{\to }A$ says that the kernel of $u$ is the zero subobject, and [L3] identifies that with $u$ being monic. This proves claim 1.

For claim 3, first assume $A\stackrel{v}{\to }B\stackrel{w}{\to }C\to 0$ is exact. Exactness at $C$ says the cokernel of $w$ is zero, so [L3] makes $w$ epic. Exactness at $B$ gives $[\mathrm{coker}(v)]=[\mathrm{coim}(w)]$ by [L2]. Because $w$ is epic, [L5] says that $w$ itself represents $\mathrm{coim}(w)$. Hence $w$ is a cokernel of $v$. Conversely, if $w$ is a cokernel of $v$, then [L4] makes $w$ epic, so exactness at $C$ follows from [L3]. Also $w$ itself represents $\mathrm{coim}(w)$, so exactness at $B$ is exactly [L2]. This proves claim 3.

For claim 2, first assume $0\to K\stackrel{u}{\to }A\stackrel{v}{\to }B$ is exact. By step 1.1, $u$ is monic. Exactness at $A$ gives $[\mathrm{im}(u)]=[\ker (v)]$ by [L2], and for a monomorphism the image representative is $u$ itself. Hence $u$ represents the same subobject as a kernel of $v$, so $u$ is a kernel of $v$. Conversely, if $u$ is a kernel of $v$, then $u$ is monic by [L4] and therefore exactness at $K$ follows from step 1.1. Since $u$ itself represents $\ker (v)$, exactness at $A$ is exactly [L2]. This proves claim 2.

If $u$ is monic and $v$ is a cokernel of $u$, then step 1.1 gives exactness of $0\to A\stackrel{u}{\to }B$, and claim 3 gives exactness of $A\stackrel{u}{\to }B\stackrel{v}{\to }C\to 0$. Hence the full sequence is short exact.

If $0\to A\stackrel{u}{\to }B\stackrel{v}{\to }C\to 0$ is short exact, then claims 2 and 3 give that $u$ is a kernel of $v$ and $v$ is a cokernel of $u$.

If $u$ is a kernel of $v$ and $v$ is epic, then claim 2 gives exactness of $0\to A\stackrel{u}{\to }B\stackrel{v}{\to }C$, and [L3] turns the epicity of $v$ into exactness at $C$. Hence the full sequence is short exact.

Steps 3.1, 2.2, and 3.2 prove claim 4.