nLab
split equalizer

A fork

$$A\xrightarrow[\quad e \quad]{}B\underoverset{\quad g \quad}{f}{\rightrightarrows}C$$

is split if it can be embedded into a diagram

$$A\xrightarrow[\quad e \quad]{\overset{\scriptsize{s}}{\begin{svg} <svg width="31" height="11" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="56258"> <g> <title>Layer 1</title> <path fill="none" stroke="#000000" d="m2.436699,8.563251c9.440479,-9.454929 17.148819,-9.609211 27.625038,0.192581" marker-start="url(#se_marker_start_svg_56258_3)" id="svg_56258_3"/> </g> <defs> <marker refY="50" refX="50" markerHeight="5" markerWidth="5" viewBox="0 0 100 100" orient="auto" markerUnits="strokeWidth" id="se_marker_start_svg_56258_3"> <path stroke-width="10" stroke="#000000" fill="#000000" d="m0,50l100,40l-30,-40l30,-40l-100,40z" id="svg_56258_7"/> </marker> </defs> </svg> \end{svg}\includegraphics[width=23]{curvedArrow1} }}B\underoverset{\quad g \quad}{\mathclap{\overset{\scriptsize{t}}{\begin{svg} <svg width="33" height="18" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="69468"> <g> <title>Layer 1</title> <path marker-start="url(#se_marker_start_svg_69468_2)" id="svg_69468_2" d="m2.12632,15.467968c13.519966,-19.398649 19.108238,-19.52702 30.125008,0.027042" stroke="#000000" fill="none"/> </g> <defs> <marker refY="50" refX="50" markerHeight="5" markerWidth="5" viewBox="0 0 100 100" orient="auto" markerUnits="strokeWidth" id="se_marker_start_svg_69468_2"> <path stroke-width="10" stroke="#000000" fill="#000000" d="m0,50l100,40l-30,-40l30,-40l-100,40z" id="svg_69468_6"/> </marker> </defs> </svg> \end{svg}\includegraphics[width=24]{curvedArrow2} }}f}{\rightrightarrows}C$$

in which $s e = id_A$, $t g = id_B$ and $t f = e s$ (we used here Leibniz order for composition of morphisms).

Surely, $f$ and $g$ can be interchanged in the definition (a matter of unimportant convention).

Every split fork is an absolute equalizer, but not conversely. See also split coequalizer.