End of a functor
= End of a functor
{title2=$\int_C T(C,C)$}
= Categorical end
{synonym}
For $T:\mathcal C^{\mathrm{op}}\times\mathcal C\to\mathcal D$, an end is a universal object $E$ with maps $e_C:E\to T(C,C)$ satisfying $T(1_C,f)e_C=T(f,1_D)e_D$ for every $f:C\to D$. Every other such family factors uniquely through $E$. For small $\mathcal C$ in a <complete category>, it is the <equalizer> of the two maps $\prod_CT(C,C)\rightrightarrows\prod_{f:C\to D}T(C,D)$.