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Let $D$ be a co-complete category and $C$ be a small category. For a functor $F:C^{op}\times C \to D$ one defines the co-end $$\int^{c\in C} F(c,c)$$ as the co-equalizer of $$\coprod_{c\to c'}F(c,c'){\longrightarrow\atop\longrightarrow}\coprod_{c\in C}F(c,c).$$ It is the indexed co-limit $\mbox{colim}_W F$ where the weight is the functor $W:C^{op}\times C \to Set$ given by $Hom(-,-)$.