While reading Fosco Loregian's "This is the (co)end, my only (co)friend" I found the following remark:
The product on the left ranges over objects in the category, but I'm confused by the so-called "double product" on the right, which seems to range over morphisms. In programmer-speak, that seems like a "type error" to me! Could someone "unpack" that part of the formula with a bit more detail?
On
Glad to see that my article is being studied :)
Another way to visualize the situation is the following, which generalizes to the enriched context:
The end of $T\colon {\bf C}^\text{op}\times {\bf C}\to {\bf D}$, for ${\bf C},{\bf D}\in {\cal V}\text{-}{\bf Cat}$ is the limit of the diagram $$ \prod_{c\in \bf C} T(c,c) \rightrightarrows \prod_{c,c'\in\bf C} T(c,c')^{{\bf C}(c,c')} $$ where $X^V$ denotes the power of an object of $\bf D$ with an object of $\cal V$. Obviously, this characterization works only when these powers exist, but this is a rather mild condition. Coends in enriched setting are defined and used in [Gra80, §2.3] and [Dub70]: I warmly invite you to use these as references.
Thanks again for your patience in reading my paper!
It is not perfectly clear to me what you have a problem with. However, writing $\mathbf{C}_0$ and $\mathbf{C}_1$ for the collections of objects and morphisms of $\mathbf{C}$, and writing $\text{dom}, \text{cod} : \mathbf{C}_1 \to \mathbf{C}_0$ for the maps sending a morphism to its domain and codomain respectively. The right hand side would be $$\prod_{f \in \mathbf{C}_1} F(\text{dom}(f),\text{cod}(f)).$$ But this could be also written as $$\prod_{(c,c')\in \mathbf{C}_1^2} \left( \prod_{f \in \hom(c,c')} F(c,c')\right).$$