TITLE: What is the functor tensor product?
QUESTION [7 upvotes]: I'm familiar with the tensor product of modules, but I've also come across functor tensor product (in emily riehls paper on homotopy limits), what are they, and how are they (if they are) related to traditional tensor products? (Emily shows that they can be defined as a particular coend, but that doesn't really provide any intuition for me).

REPLY [3 votes]: Familiar examples of modules can be expressed in terms of categories. For example:

If $\mathcal{G}$ is a one-object groupoid, then functors $\mathcal{G}^\circ \to \mathbf{Set}$ are the same thing as right actions of the group $\hom_{\mathcal{G}}(*,*)$ on sets.
If $\mathcal{R}$ is a one-object additive category, then additive functors $\mathcal{R} \to \mathbf{Ab}$ are the same thing as left modules over the ring $\hom_{\mathcal{R}}(*,*)$

Generalizing, we can view any functor $\mathcal{C} \to \mathcal{D}$ as a left action of $\mathcal{C}$ (on objects of $\mathcal{D}$). Similarly, functors $\mathcal{C}^\circ \to \mathcal{D}$ are right actions. The most common example is actions on sets; i.e. $\mathcal{D} = \mathbf{Set}$. 
Let there be a presheaf $P : \mathcal{C}^\circ \to \mathbf{Set}$, a functor $F : \mathcal{C} \to \mathbf{Set}$, and a set $S$.
As with other sorts of modules, there is a "hom-tensor" adjunction: there is a functor $\otimes_\mathcal{C} : \mathbf{Set}^{\mathcal{C}^\circ} \times \mathbf{Set}^{\mathcal{C}} \to \mathbf{Set} $ such that there is a natural bijection
$$\hom_\mathbf{Set}(P \otimes_\mathcal{C} F, S) \cong \hom_{\mathbf{Set}^{\mathcal{C}^\circ}}(P, \hom_{\mathbf{Set}}(F-, S))$$
We can also characterize the tensor product by the facts. Let $\mathbf{y} : \mathcal{C} \to \mathbf{Set}^{\mathcal{C}^\circ}$ be the yoneda embedding. Then,

$P \otimes_\mathcal{C} F$ preserves colimits in both variables
There is a natural bijection $\mathbf{y}(C) \otimes_\mathcal{C} F \cong F(C)$

One of the important intuitions about presheaves are that they are formal colimit diagrams. If $P$ is written as a colimit of representables, then we can see that
$$ P \otimes_\mathcal{C} F \cong \left( \operatorname{colim}_j \mathbf{y}(C_j) \right) \otimes_\mathcal{C} F  \cong \operatorname{colim}_j F(C_j) $$
This meshes with other tensor product intuitions; for example, you can imagine $P$ as a "free" colimit, and $- \otimes_\mathcal{C} F$ specializes the formal generators $\mathbf{y}(C_j)$ to the special values $F(C_j)$.
The story generalizes almost unchanged to functors $F : \mathcal{C} \to \mathcal{M}$ for a locally small, cocomplete $\mathcal{M}$. You still have a functor $\otimes_\mathcal{C} : \mathbf{Set}^{\mathcal{C}^\circ} \times \mathcal{M}^{\mathcal{C}} \to \mathbf{Set} $  and a hom-tensor adjunction
$$\hom_\mathcal{M}(P \otimes_\mathcal{C} F, M) \cong \hom_{\mathbf{Set}^{\mathcal{C}^\circ}}(P, \hom_{\mathcal{M}}(F-, M))$$
you still have the properties

$P \otimes_\mathcal{C} F$ preserves colimits in both variables
There is a natural bijection $\mathbf{y}(C) \otimes_\mathcal{C} F \cong F(C)$

and $P \otimes_\mathcal{C} F$ can be interpreted as a colimit computation in $\mathcal{M}$.
These stories can be generalized to the enriched setting; my impression that it's easier to see how various constructions should go when they're phrased in terms of homs and tensors rather than in other terms.