In the [paper](http://www.ams.org/journals/tran/1997-349-01/S0002-9947-97-01752-2/S0002-9947-97-01752-2.pdf)



> 
> Cordier, Jean-Marc, and Timothy Porter. "Homotopy coherent category theory." Transactions of the American Mathematical Society 349.1 (1997): 1-54.
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the authors define a notion of *coherent co/end* $\oint T(a,a)$ as
$$
\int\_{a',a''} T(a',a'')^{\delta A(a',a'')}
$$
where $\delta A(a',a'')$ is a suitable "resolution" of the hom functor (linked to the bar construction; follow §1, and in particular the paragraphs above Definition 1.1).


The main claim of this paper is that category theory can be rewritten in a homotopy coherent fashion, building on this definition of homotopy coherent co/end. It is obvious why I like this point of view :-)


One of the first interesting claims is the following: fix a simplicial functor $F : A \to B$ between simplicial categories, and define
$$
\begin{gather}
\overline F = \{\delta A(a,-),F\} = \oint\_{x:A} Fx^{A(a,x)}\\
\underline F = \delta A(-,a)\otimes F = \oint^{x:A} Fx \otimes A(x,a)
\end{gather}
$$
This is the counterpart of the "ninja Yoneda lemma" in covariant and contravariant form.


Coend-fu now gives that "underlines and overlines absorb coherence" in that
$$
\text{CohNat}(F,G) \cong \text{Nat}(F, \overline G)\cong \text{Nat}(\underline F,G)
$$
This gives natural transformations
$$ \eta\_F : F \Rightarrow \overline F \qquad\qquad \eta^F : \underline F \Rightarrow F$$
as the images of the identity coherent natural transformation of $F$. The authors then prove that these maps are levelwise homotopy equivalences (Prop. 3.4).



> 
> Are $\overline F, \underline F$ replacements for $F$ in suitable model structures on categories of functors? Is $\overline F$ fibrant if $F \in [A, {\bf sSet}]\_\text{inj}$? Is $\underline F$ cofibrant if $F \in [A, {\bf sSet}]\_\text{proj}$?
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Co/fibrant replacements via coend calculus
Addressing this sort of question was one of the main goals of [math/0610194](https://arxiv.org/abs/math/0610194). The best answer I was able to give is that if $B$ has a suitable model structure with respect to which $F$ is *objectwise* fibrant (resp. cofibrant), then $\overline{F}$ (resp. $\underline{F}$) belongs to a "right (resp. left) deformation retract" (an abstraction of the notion of fibrant (resp. cofibrant) replacement) that is suitable for constructing derived functors of functors such as homotopy limits (resp. colimits) and also the homotopy category of the functor category (which therefore involves homotopy coherent transformations).


A closely related perspective that you may also be interested in can be found in [this paper](http://www.sciencedirect.com/science/article/pii/S0022404909002412) by Gambino: the colimit of $\underline{F}$ is equivalently the colimit of $F$ weighted by a projective-cofibrant replacement of the terminal weight.


