Kernel Pair

The kernel pair of a morphism f : X Y : 𝑓 𝑋 𝑌 f:{{X}\to{Y}} is the Pullback

Y×XYsubscript𝑋𝑌𝑌{{Y\times_{X}Y}}Y𝑌{{Y}}Y𝑌{{Y}}X𝑋{{X}}π1subscript𝜋1\scriptstyle{\pi_{1}}π2subscript𝜋2\scriptstyle{\pi_{2}}f𝑓\scriptstyle{f}f𝑓\scriptstyle{f}\scriptstyle{\lrcorner}

of f 𝑓 f along itself.