Let $D: \mathcal{C} \to \mathbf{Set}$ be a diagram of sets, then we can obtain the colimit of $D$ as the set of connected components of the category of elements of $F$, which we denote by $\mathrm{el}(F)$. The category $\mathrm{el}(F)$ in turn is the oplax colimit of the composition of $F$ and the inclusion $\mathbf{Set} \hookrightarrow \mathbf{Cat}$.
Now let us consider a functor $F: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathbf{Set}$, then the coend of $F$ can again be constructed as the set of connected components of a category $\mathrm{el}^\wedge(F)$ which we now describe:

• The objects are pairs $(X,x)$ with $X \in \mathcal{C}$ and $x \in F(X,X)$.
• A morphism $(X,x) \to (Y,y)$ is given by a pair $(f,a)$, where $f: X \to Y$ is morphism in $\mathcal{C}$, and $a \in F(X,Y)$ such that $f_*(x) = f^*(y) = a$.
• Finally composition of two morphisms $(X,x) \xrightarrow{(f,a)} (Y,y) \xrightarrow{(g,b)} (Z,z)$ is given by $(g \circ f, g_*(a)) = (g \circ f, f^*(b))$.

Does the category $\mathrm{el}^\wedge F$ satisfy a similar universal property as $\mathrm{el}F$?

The above description of the colimit of $D$ feels very natural to me, and I was hoping that this perspective may shed some light on the nature of coends*.

Finally I would like to note that it may be that there are other categories $\mathcal{E}$ than $\mathrm{el}^\wedge F$ such that $\pi_0(\mathcal{E})$ corresponds to the coend of $F$, which are more natural than my construction. If this is the case, I would be grateful for these to be pointed out.

_{*It is like first taking the quotient of a set by a discrete group in the $(\infty,1)$-category of spaces (or in this case equivalently in the $(2,1)$-category of groupoids) and then applying the left adjoint to $\mathbf{Set} \hookrightarrow \mathbf{Spaces}$ (resp. $\mathbf{Set} \hookrightarrow \mathbf{Groupoids}$).}