Factorization systems

If (a category $\calE$) has (equalizers and coequalizers, kernel pairs and cokernel pairs)
then for each (arrow $\arrowf$ in $\calE$) we have (the commutative diagram in $\calE$) which appears in (the center of the display below);
to its left and right are (2-diagrams in $\CAT$) showing parts of (the adjunctions determined by those limit and colimits in $\calE$) : $\Newextarrow{\xrightrightarrows}{5,5}{0x21C9} \Newextarrow{\xrightarrowtail}{5,5}{0x21A3}$ \[\mkern-3em \begin{array}{} && \CAT &&&&&&& \calE &&&&&&& \CAT \\ \\ \calE\downarrow\objY && \longrightarrow && \calE\downarrow\objY & \mkern3em & \arrowf\Kp & \xrightrightarrows{\textstyle\arrowf\kp} & \objX & \xtwoheadrightarrow[\textstyle \arrowf\kp\coequ]{\href{https://ncatlab.org/nlab/show/regular+epimorphism}{\text{regular epi.}}} & \arrowf\kp\Coequ \rlap{{} \equiv \arrowf\Coim} & && &&& \calE\downdownarrows\setX \\ & \llap\cokp \searrow & \Bigg\Downarrow\rlap\eta & \nearrow\rlap\equ &&& && \llap{\arrowf\eta}\Bigg\downarrow & \llap\arrowf \searrow & \Bigg\downarrow\rlap{\arrowf\epsilon} &&&&& \llap\kp\nearrow & \Bigg\Downarrow\rlap\epsilon & \searrow\rlap\coequ \\ && \setY\downdownarrows\calE &&&& && \llap{\arrowf\Im \equiv {}} \arrowf\cokp\Equ & \xrightarrow[\href{https://ncatlab.org/nlab/show/regular+monomorphism}{\text{regular mono.}}]{\textstyle\arrowf\cokp\equ} & \objY & \xrightrightarrows[\textstyle\arrowf\cokp]{} & \arrowf\Cokp & \mkern3em & \setX\downarrow\calE && \longrightarrow && \setX\downarrow\calE \\ \end{array}\] Further, (“the diagonal fill-in property”) there is one and only one
[(diagonal arrow from $\arrowf\Coim$ to $\arrowf\Im$) which makes (both of the triangles of which it is an edge) commute].

All of this follows easily from (the properties of the limits and colimits which are mentioned in the diagram).

In many cases the diagonal fill-in is an isomorphism.
The paradigmatic example is when $\calE=\Set$.
Then (the unique diagonal fill-in) is
[the canonical bijection between (the set of blocks in (the partition of $\setX$ determined by $\functionf$), i.e., the fibers of $\functionf$) and (the image of $\functionf$ as a subset of $\setY$)].
Concrete examples illustrating how this works, in the familiar case $\calE=\Set$,
are given in the posts "The parts of a function" and "Classifying functions by their parts",
using a slightly different language aimed at readers more familiar with set theory than category theory.