Co-ends as a trace operation on profunctors The n-lab site on profunctors (http://ncatlab.org/nlab/show/profunctor) describes profunctor composition as using a co-end to "trace out" the connecting variable:
$F\circ G := \int^{d\in D} F(-, d) \times G(d, -)$
Naturally, one could picture for more general profunctors, $\psi : A \times B^{op} \times C \times D \rightarrow Set$, $\phi : B \times E^{op} \rightarrow Set$ a more generalised composition as co-end'ing together an input from one with an output from the other:
$\xi := \int^{b \in B} \psi(-,b,-,-) \times \phi(b,-)$
This then looks a lot like contraction of tensors:
$\xi_{e}^{a,c,d} := \sum_{b\in B} \left( \psi_{b}^{a,c,d} \phi_{e}^{b} \right)$
Even in the use of the language "trace out" (and the fact that both operations form abstract trace operations $Tr(-)$ in their respective traced monoidal categories), this analogy seems to be implied. This also seems to be a useful way to think about profunctor composition, and it appears quite feasible that tensor contraction could be described as a co-end of suitably-enriched profunctors. However, it doesn't seem obvious how to go about unifying these two operations. So, my question is:

To what extent can the analogy between tensor contraction and profunctor composition be made precise?

 A: The connection can be made precise via the notion of compact closure for symmetric monoidal structures. 
Recall that a symmetric monoidal category $C$ is compact closed if every object $c$ has a right adjoint $c^\ast$, meaning that there are unit and counit arrows $\eta: I \to c^\ast \otimes c$ and $\varepsilon: c \otimes c^\ast \to I$ satisfying triangular equations ($I$ is the monoidal unit; the idea is that $c^\ast \otimes -$ is right adjoint to $c \otimes -$). The classical example is of course the category of finite-dimensional vector spaces: the counit $V \otimes V^\ast \to k$ is given by evaluation, and the unit $k \to V^\ast \otimes V$ takes the unit element $1 \in k$ to $\sum_i f^i \otimes e_i$ for any chosen basis $e_i$ and dual basis $f^i$. 
In such a situation, there are various equivalent ways of considering morphisms $f: b \to c$. By the adjunction, they are in natural bijection with morphisms $I \to b^\ast \otimes c$. (I'll call this the 'right' picture.) By taking advantage of the symmetry of the tensor, it's also true that one can switch things around and give an adjunction $c^\ast \dashv c$, and thus morphisms $f: b \to c$ will also be in natural bijection with morphisms $b \otimes c^\ast \to I$ (the 'left' picture). Also in this situation, one can define an abstract trace of an endomorphism $f: b \to b$ by the formula 
$$Tr(f) = (I \stackrel{unit}{\to} b \otimes b^\ast \stackrel{f \otimes 1_{b^\ast}}{\to} b \otimes b^\ast \stackrel{counit}{\to} I)$$ 
which returns the classical trace for endomorphisms on a finite-dimensional vector space. Continuing with this, the composition of two morphisms in the right picture, say $f: I \to b^\ast \otimes c$ and $g: I \to c^\ast \otimes d$, is obtained by 'tracing out': 
$$I \stackrel{f \otimes g}{\to} b^\ast \otimes c \otimes c^\ast \otimes d \stackrel{1_{b^\ast} \otimes \varepsilon \otimes 1_d}{\to} b^\ast \otimes d$$ 
which is to say composing with the counit in a tensor sandwich. 
Now, the bicategory of small categories and profunctors is a compact closed bicategory, meaning that it is a symmetric monoidal bicategory (the tensor being given at the object level by taking cartesian products), and every object $C$ has a right biadjoint, which turns out to be $C^{op}$. If we think of a profunctor from $C$ to $D$ as essentially the same thing as a cocontinuous functor 
$$Set^{C^{op}} \to Set^{D^{op}},$$
then the unit $1 \to C^{op} \otimes C$ in $Prof$ corresponds to the unique (up to iso) cocontinuous functor 
$$Set \to Set^{C^{op} \times C}$$ 
that sends the terminal object $1$ to $\hom_C$. The counit $C \otimes C^{op} \to 1$ corresponds to a cocontinuous functor 
$$Set^{C \times C^{op}} \to Set$$ 
which sends a functor $F: C \times C^{op} \to Set$ to the coend $\int^c F(c, c)$. Thus, composition of profunctors in the right picture will involve a tracing out by applying a coend operation to the middle two factors. 
