TITLE: Why do we denote (co)ends with integral notation (beyond Fubini's Theorem)?
QUESTION [19 upvotes]: I know that (co)ends (i.e. universal wedges) follow Fubini-like relation, i.e.
$$ \int_{\langle c,d\rangle} F(c,d,c,d) \cong \int_c\int_d F(c,c,d,d) \cong \int_d\int_c F(c,c,d,d) $$
where we regard $F$ as both a functor $(C\times D)^{op}\times (C\times D)\to X$ and a functor $C^{op}\times C \times D^{op} \times D \to X$. 
Is there any other reason we use this notation? E.g. is their formulation as a limit in some way analogous to Riemann sums, do they have some kind of linearity relation, etc?

REPLY [5 votes]: Since coends commute with colimits, in particular coproducts, we have some kind of "linearity"
$$\int^i \bigl(X(i,i) \oplus Y(i,i)\bigr) \cong \int^i X(i,i) \oplus  \int^i Y(i,i).$$
Every functor $F : C \to D$ into a cocomplete category $D$ may be decomposed as
$$F \cong \int^{x \in C} F(x) \otimes \hom(x,-),$$
this is basically a reformulation of the Yoneda Lemma. Here, $\otimes$ denotes the copower $D \times \mathsf{Set} \to D$.
This looks very similar to the equation (in fact, this is how I remember this)
$$f(y) = \int_{x \in \Omega} f(x) \cdot d \delta_y(x),$$
where $\delta_y$ denotes the dirac measure concentrated at $y \in \Omega$ and $f : \Omega \to \mathbb{R}$ is a measurable function, as well as to the equation
$$\mu(A) = \int_{x \in \Omega} d \mu(x) \cdot \chi_A(x)$$
for measures $\mu$ on $\Omega$.
However, the analogy between coends and integrals, in particular between coproducts and sums doesn't go too far: The sum of real numbers has no analogue of a universal property. There are no analogues of coproduct inclusions. In category theory there are no nontrivial objects which are invertible with respect to coproducts. And sums in analysis are not really dual to products. Finally notice that Fubini's theorem in category theory is a formal consequence of the definitions, whereas Fubini's theorem in measure theory is not.