Coend of a functor
= Coend of a functor
{title2=$\int^C T(C,C)$}
= Categorical coend
{synonym}
The dual universal family is a coend $Q$ with maps $i_C:T(C,C)\to Q$ satisfying $i_CT(f,1_C)=i_DT(1_D,f)$ on $T(D,C)$. Every compatible family to another object factors uniquely through $Q$. For small $\mathcal C$ in a cocomplete category, it is the <coequalizer> $\coprod_{f:C\to D}T(D,C)\rightrightarrows\coprod_CT(C,C)\to Q$.