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 $Hom(-,-)$.

I have two strongly related questions regarding this definition. First, what's the intuition behind this construction? Can I think of it as a kind of ''fattened'' colimit? Second, why is the integral sign used for this? Can ordinary integration be related to this construction?