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?



Why do we denote (co)ends with integral notation (beyond Fubini's Theorem)?
It's perhaps not *great* notation, but some of the most common types of coends arising in practice (namely, weighted colimits) can be thought of roughly as "categorified weighted sums". 


In enriched category theory (over a complete, cocomplete symmetric monoidal closed category $V$), a *weight* consists of a small $V$-category $J$ and a $V$-functor $W: J^{op} \to V$, and the *weighted colimit* of a functor $F: J \to C$ over the weight $W$ is an object $c$ together with a $V$-natural isomorphism 


$$C(c, d) \cong V^{J^{op}}(W-, V^J(F-, d))$$ 


(where as usual in this subject, $\hom\_C(d, d)$ is abbreviated to $C(c, d)$). The typical way this colimit is presented, for a $V$-cocomplete category $C$, is by means of (as always, $V$-enriched) coends and tensors (where we denote tensors with a dot symbol $\cdot$):


$$c = \int^{j: J} W j \cdot F j.$$ 


Let's consider the most vanilla case possible, where $V$ is just $\text{Set}$ and $J$ is small discrete (in other words, a set). A weight in this case just assigns a set $W(j)$ to each element $j \in J$; the $W(j)$ might be finite or infinite, but no matter. Then this coend formula just amounts to a coproduct 


$$c = \sum\_{j \in J} W(j) \cdot F(j)$$ 


where $W(j) \cdot F(j)$ means we are taking a coproduct of a bunch of copies of $F(j)$, one for each element $x \in W(j)$. In other words, the weighted colimit is a categorified weighted sum of $F$, somewhat analogous to weighted sums in analysis (e.g., integration of a function $f$ against a measure $d\mu = w(x) dx$ as in Lebesgue-Stieltjes integration). 


The more general weighted colimits are not coproducts of course, but they are related. Again, to take the vanilla case $V = \text{Set}$, the coend formula amounts to a coequalizer appearing in a diagram: 


$$\sum\_{j, k \in J} W(j) \cdot J(k, j) \cdot F(k) \rightrightarrows \sum\_j W(j) \cdot F(j) \to \int^j W(j) \cdot F(j)$$ 


where the two parallel arrows are induced by canonical maps $W(j) \cdot J(k, j) \to W(k)$ and $J(k, j) \cdot F(k) \to F(j)$, which in turn owe their presence to the functorial structure of $W$ and $F$. 


