End and coend

Remarks

The end is written $\int_{c} S(c, c)$ and the coend $\int^{c} S(c, c)$. Both are unique up to a unique isomorphism. For $\mathcal{C}$ small and $\mathcal{D}$ complete, the end is the equalizer of two maps $\prod_c S(c, c) \rightrightarrows \prod_{f \colon c \to c'} S(c, c')$.

Example: let $F, G \colon \mathcal{C} \to \mathcal{E}$ be functors with $\mathcal{C}$ small and $\mathcal{E}$ locally small, and let $S(c, c') = \Hom_{\mathcal{E}}(F c, G c')$ in $\mathbf{Set}$. A wedge from a one-point set is a family $\alpha_c \colon F c \to G c$ with $G f \circ \alpha_c = \alpha_{c'} \circ F f$, that is, a natural transformation. So $\int_c \Hom(F c, G c)$ is the set of natural transformations $F \Rightarrow G$.

Non-example: in the same setting, the set $\prod_c \Hom(F c, G c)$ of all families, with its projections, is in general not a wedge. The wedge condition would require every family $(\alpha_c)$ to be natural. It fails as soon as some family has $G f \circ \alpha_c \neq \alpha_{c'} \circ F f$ for some $f \colon c \to c'$.