Kelly ECT displayed formulae and diagrams

PRELIMINARY DRAFT --- CUSTOMIZED, NON-STANDARD

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. 

\begin{align} \boxed{\int_{A\in\calA} \hom A T A} & {\mkern2em} \xrightarrow{\text{equalizer}} {} & \prod_{A\in\calA} \hom A T A  & {\mkern2em} \mathop\rightrightarrows\limits^\rho_\sigma {} & \prod_{A,A'\in\calA} [ \hom A \calA {A'}, \hom A T {A'} ] \\ & \mkern2em  \llap{\boxed{\lambda_A}} \searrow & \downarrow \rlap{\text{proj}_A} \\ && \hom A T A \tag{2.2} \end{align}

\begin{gather} \Big[X, \int_{A\in\calA} \hom A T A \Big] \xrightarrow{\textstyle [X,\lambda_A]} [X, \hom A T A] \text{ makes its source the end of lts targets, inducing}\\ \Big[ X, \int_{A\in\calA} \hom A T A \Big] \cong \int_{A\in\calA} [X, \hom A T A ] \tag{2.3} \end{gather}

\[ \int_{(A,B)\in\calA\tensor\calB} \hom {A,B} T {A,B} \cong \int_{B\in\calB} \int_{A\in\calA} \hom {A,B} T {A,B} \tag{2.8} \]

\begin{align} \boxed{\hom T {[\calA,\calB]} S} & \mkern2em \cong & \int_{A\in\calA} \hom {AT} \calB {AS} & {\mkern2em} \xrightarrow{\text{equalizer}} & \prod_{A\in\calA} \hom {AT} \calB {AS}  & {\mkern2em} \mathop\rightrightarrows\limits^\rho_\sigma  & \prod_{A,A'\in\calA} [ \hom A \calA {A'} , \hom {AT} \calB {A'S}  ] \\ &  \mkern2em \llap{\boxed{E_A}} \searrow & \downarrow \rlap{\text{proj}_A}  \mkern1em \\ && \hom {AT} \calB {AS} \tag{2.10} \end{align}

\[\big[ X, \hom T {[\calA,\calV]} S \big] \cong \hom T {[\calA,\calV]} {[X,S]} \tag{2.13} \]

Given a $\calV$-functor $\boxed{F:\calA\to\calV}$ and an object $\boxed K$ of $\calA$ as in 1.9, 

we have the map (arrow in $\calV_0$) $\hom K F A : \hom K \calA A \to [F_K,F_A]$, 

which is.$\calV$-natural in $A$ by 1.8(b).

The transform \[ \phi_A : F_K \to [\hom K \calA A, F_A] \tag{2.30} \] of $\hom K F A$ under the (symmetry) adjunction $\hom X {\calV_0} {[Y,Z]} \cong \hom Y {\calV_0} {[X,Z]}$

is ($\calV$-natural in $A$) by 1.8(m).

(The stronger Yoneda lemma) is 

the assertion that (2.30) expresses $F_K$ as the end $ \boxed{\textstyle \int_A [\hom K \calA A, F_A]}$, 

so that we have an isomorphism (natural in $K$) \[\boxed{ \phi : F_K \cong \hom {\hom K \calA -} {[\calA,\calV]} F }. \tag{2.31} \]

$F_K$ was a single-variable functor; Kelly generalizes to a functor $\hom C P K$ of two variables in the (natural in the "extra" variable $C$, as well as in $K$):

\[ \boxed{ \phi = \hom C \phi K : \hom C P K  \cong \hom {\hom K \calA -} {[\calA,\calV]} {\hom C P -} } \tag{2.33} \]

\begin{align} \boxed{\hom B \calB {\{\hom K \calK - , G\}}} & \mkern2em \mathop\cong\limits^{(3.1)} & \hom {\hom K \calK -} {[\calK,\calV]} {\hom B \calB G} && \Rule 2px 3ex 1ex && \boxed{\hom {(\hom - \calK K * G)} \calB B} & \mkern2em \mathop\cong\limits^{(3.5)} & \hom {\hom - \calK K} {[\calK\op,\calV]} {\hom G \calB B}  \\ &&  {\wr\Vert} \rlap {(2.33) \text{ Yoneda}} \mkern2em && \Rule 2px 3ex 1ex && && \mkern3em {\wr\Vert} \rlap {(2.33) \text{ Yoneda}} \mkern2em \\ &&  \boxed{\hom B \calB {G_K}} \mkern2em && \Rule 2px 3ex 1ex && && \boxed{\hom {G_K} \calB B} \mkern2em \tag{3.10} \end{align}

\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}