Kernel of an arrow that factors through a monic? Suppose an arrow $A\overset{f}{\rightarrow}B$ factors as $A\overset{q}{\rightarrow} J \overset{j}{\rightarrowtail}B$. When does $\ker f=\ker q$ and how can I prove it?--prophetes.ai

Kernel of an arrow that factors through a monic? Suppose an arrow $A\overset{f}{\rightarrow}B$ factors as $A\overset{q}{\rightarrow} J \overset{j}{\rightarrowtail}B$. When does $\ker f=\ker q$ and how can I prove it?

Always. The kernel of $f$ represents maps $z$ into $A$ which are killed by composition with $f$; but if $fz=jqz=0,$ then $qz=0$ and $z$ factors also through the kernel of $q$. In the other direction, if $qz=0$ then certainly $fz=0$.