This is just a beginning on that project.
Currently only a few selected formulae and diagt are listed. Also some of the Kelly's words have been included.
The formulae are not exactly what Kelly wrote in ECT. I have transformed them somewhat.
∫A∈AATAequalizer→∏A∈AATAρ⇉σ∏A,A′∈A[AAA′,ATA′]λA↘↓projAATA
[X,∫A∈AATA][X,λA]→[X,ATA] makes its source the end of lts targets, inducing[X,∫A∈AATA]≅∫A∈A[X,ATA]
∫(A,B)∈A⊗BA,BTA,B≅∫B∈B∫A∈AA,BTA,B
T[A,B]S≅∫A∈AATBASequalizer→∏A∈AATBASρ⇉σ∏A,A′∈A[AAA′,ATBA′S]EA↘↓projAATBAS
[X,T[A,V]S]≅T[A,V][X,S]
Given a V-functor F:A→V and an object K of A as in 1.9,
we have the map (arrow in V0) KFA:KAA→[FK,FA],
which is.V-natural in A by 1.8(b).
The transform ϕA:FK→[KAA,FA] of KFA under the (symmetry) adjunction XV0[Y,Z]≅YV0[X,Z]
is (V-natural in A) by 1.8(m).
(The stronger Yoneda lemma) is
the assertion that (2.30) expresses FK as the end ∫A[KAA,FA],
so that we have an isomorphism (natural in K) ϕ:FK≅KA−[A,V]F.
FK was a single-variable functor; Kelly generalizes to a functor CPK of two variables in the (natural in the "extra" variable C, as well as in K):
ϕ=CϕK:CPK≅KA−[A,V]CP−
BB{KK−,G}(3.1)≅KK−[K,V]BBG(−KK∗G)BB(3.5)≅−KK[Kop,V]GBB≀‖
\begin{align} \hom C Q {\{F,G\}} & \mathop\cong\limits^{Q \text{ preserves}}_{\text{limit}} & {\{F,\hom C Q G\}} & \mathop\cong\limits^{ (3.7)} & \hom F {[\calK,\calV]} {\hom C Q G} \\ \llap{\text{defn of $T$ }} {\wr\Vert} \\ \boxed{\hom C \calC {\{F,G\}T}} &&&& {\wr\Vert} \rlap{\text{ defn of $T$}} \\ \\ \boxed{\hom C \calC {\{F,GT\}}} && \mathop\cong\limits^{(3.1)} && \hom F {[\calK,\calV]} {\hom C \calC {GT}} \tag{3.15} \end{align}
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