The Stacks project

Proof. Suppose the diagram is cartesian. Let $e:\mathop{\mathrm{Ker}}(f)\to \mathop{\mathrm{Ker}}(k)$ be induced by $g$. Let $i:\mathop{\mathrm{Ker}}(f)\to w$ and $j:\mathop{\mathrm{Ker}}(k)\to x$ be the canonical injections. There exists $t:\mathop{\mathrm{Ker}}(k)\to w$ with $f\circ t=0$ and $g\circ t=j$. Hence, there exists $u:\mathop{\mathrm{Ker}}(k)\to \mathop{\mathrm{Ker}}(f)$ with $i\circ u=t$. It follows $g\circ i\circ u\circ e=g\circ t\circ e=j\circ e=g\circ i$ and $f\circ i\circ u\circ e=0=f\circ i$, hence $i\circ u\circ e=i$. Since $i$ is a monomorphism this implies $u\circ e=\text{id}_{\mathop{\mathrm{Ker}}(f)}$. Furthermore, we have $j\circ e\circ u=g\circ i\circ u=g\circ t=j$. Since $j$ is a monomorphism this implies $e\circ u=\text{id}_{\mathop{\mathrm{Ker}}(k)}$. This proves (1). Now, (2) follows by duality. $\square$