The category structures on 2

• $\cattwo$ as a metacategory (CWM 1.1)
• $\cattwo$ as an internal category in $\Set$, i.e., $\cattwo\in\Cat(\Set)$ (CWM 1.2)
• $\cattwo$ as a category enriched in $\Set$, i.e., as a $\Set$-category, i.e., $\cattwo\in\Set$-$\Cat$ (CWM 1.8, 7.7)
• $\bftwo$ as a category enriched in itself, i.e., as a $\bftwo$-category, i.e., $\bftwo\in\bftwo$-$\Cat$ (BCECT 1.6.3)
• $\Set$ and $\cattwo$ as $\text{ccccc}$s - complete and cocomplete cartesian closed categories
• Relations between the categories

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$\cattwo$ as a metacategory (CWM 1.1)

(The metacategory $\cattwo$) has data consisting of:
two objects, $\bot$ and $\top$, and three arrows, $1_\bot:\bot\to\bot$, ${\lt}:\bot\to\top$, and $1_\top:\top\to\top$
(${\lt}:\bot\to\top$ is of course usually denoted $\bot\lt\top$).
(Its identity arrows) are evident, and (its composition operation) is defined by (the unit axiom).
Note that no mention of sets has been made.

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$\cattwo$ as an internal category in $\Set$, i.e., $\cattwo\in\Cat(\Set)$ (CWM 1.2)

(The internal category $\cattwo$ in $\Set$) has data consisting of
a set of objects $\boxed{\cattwo_0 = \{\bot,\top\}}$ and
a set of arrows $\boxed{\cattwo_1 = \{1_\bot,{\lt},1_\top\}}$,
together with (source, target, identity, and composition functions) which implement (the operations defined for the metacategory $\cattwo$).
The only difference between the definition of $\cattwo$ (as a metacategory) and (as an internal category in $\Set$) is (the explicit mention of sets and functions).

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$\cattwo$ as a category enriched in $\Set$, i.e., as a $\Set$-category, i.e., $\cattwo\in\Set$-$\Cat$ (CWM 1.8, 7.7)

Again, we have the set of objects $\{\bot,\top\}$, now denoted by $\boxed{\ob\cattwo}$ (because the subscript $()_0$ will soon be given another meaning).
We also have a hom-function $\boxed{ \hom - \cattwo - : \ob\cattwo \times \ob\cattwo \to \Set}$,
taking each (pair of objects) to {the set of arrows from (the first, source, object) to (the second, target, object)}.
We represent the hom-function as a table:

target	$\top$		$\{\lt\}$	$\{1_\top\}$
	$\bot$		$\{1_\bot\} $	$\emptyset$
$\hom {\langle\text{source}\rangle} {\cattwo} {\langle\text{target}\rangle}$		$\bot$	$\top$
		source

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$\bftwo$ as a category enriched in itself, i.e., as a $\bftwo$-category, i.e., $\bftwo\in\bftwo$-$\Cat$ (BCECT 1.6.3)

(The internal hom for $\bftwo$) is shown in (the center of the display below), where it can be easily compared to (some related functions).

The $\Set$-valued hom-function (the external hom) for the $\Set$-category $\cattwo$ target $\top$ $\{ \bot \xrightarrow{\textstyle \lt} \top \}$ $\{ \top \xrightarrow{\textstyle 1_\top} \top \}$ $\bot$ $\{ \bot \xrightarrow{\textstyle 1_\bot} \bot\}$ $\emptyset$ $\hom {\langle\text{source}\rangle} {\cattwo} {\langle\text{target}\rangle}$ $\bot$ $\top$ source	The $\bftwo$-valued hom-function (the internal hom) for the $\bftwo$-category $\bftwo$ target $\top$ $\top$ $\top$ $\bot$ $\top$ $\bot$ $\hom {\langle\text{source}\rangle} \bftwo {\langle\text{target}\rangle}$ or $[\langle\text{source}\rangle,\langle\text{target}\rangle]$ $\bot$ $\top$ source	The truth-table for implication ($\Rightarrow$) consequent $\top$ $\top$ $\top$ $\bot$ $\top$ $\bot$ $\langle\text{ant.}\rangle \Rightarrow \langle\text{cons.}\rangle$ $\bot$ $\top$ antecedent
	The $\Set$-valued hom-function for the full subcategory $\cal T$ (for “tiny”) determined by $\emptyset$ and $\{\emptyset\}$ of the $\Set$-category $\Set$ target $\{\emptyset\}$ $\big\{\emptyset\to\{\emptyset\}\big\}$ $\big\{\{\emptyset\} \xrightarrow{\textstyle 1_{\{\emptyset\}} } \{\emptyset\}\big\}$ $\emptyset$ $\big\{\emptyset \xrightarrow{\textstyle 1_\emptyset } \emptyset\big\}$ $\emptyset$ $\hom {\langle\text{source}\rangle} \Set {\langle\text{target}\rangle}$ or $[\langle\text{source}\rangle,\langle\text{target}\rangle]$ $\emptyset$ $\{\emptyset\}$ source

We can compare (the two $\Set$-categories, $\cattwo$ and $\Set$) to (the $\bftwo$-category $\bftwo$) with (the following two diagrams).
The labels above the arrows are the names of the arrows in the $\Set$-categories
(for the last arrow on each line, there is no label above, indicating that there is no such arrow in the $\Set$-category; the $\Set$ arrow $\emptyset\to\{\emptyset\}$ exists, but is here unnamed);
labels below arrows indicate the value of $\big( \hom {\langle\text{source}\rangle} \bftwo {\langle\text{target}\rangle} = [\langle\text{source}\rangle,\langle\text{target}\rangle] \big)$ for the indicated source and target. \[\begin{array}{ccccccccccl} \bot & \xrightarrow[\textstyle \top]{\textstyle 1_\bot} & \bot & \xrightarrow[\textstyle \top]{\textstyle \lt} & \top & \xrightarrow[\textstyle \top]{\textstyle 1_\top} & \top & \xrightarrow[\textstyle \bot]{} & \bot & \mkern6em & \cattwo \text{ above, } \bftwo \text{ below} \\ \\ \emptyset & \xrightarrow{\textstyle 1_\emptyset} & \emptyset & \xrightarrow{} & \{\emptyset\} & \xrightarrow{\textstyle 1_{\{\emptyset\}}} & \{\emptyset\} & \xrightarrow{} & \emptyset && \Set \text{ above} \\ \end{array}\]

There is a lot of information packed into those hom-tables. For example:
[That (the edge corresponding to ($\langle\text{source}\rangle=\bot$)) is all $\top$] is equivalent to ($\bot$ being initial in $\bftwo_0$).
[That (the edge corresponding to ($\langle\text{target}\rangle=\top$)) is all $\top$] is equivalent to ($\top$ being terminal in $\bftwo_0$).
[That (the edge corresponding to ($\langle\text{source}\rangle=\top$)) is identical to its input] is equivalent to ($[\top,-] \xlongequal{\textstyle i} 1_\bftwo$). (The letter $i$ comes from a generalization in the paper [CC].)
[That (the edge corresponding to ($\langle\text{target}\rangle=\bot$)) interchanges truth values] is equivalent to ($[-,\bot] = \neg$ (negation)).

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The subscript $()_0$ changes meaning when going between (internal category theory) and (enriched category theory)

On the one hand, (an internal category $\catA$ in a category $\calE$) can be viewed as (a truncated simplicial object in $\calE$),
with $\catA_0$ = the object of objects, $\catA_1$ = the object of arrows, $\catA_2$ = the object of composable pairs of arrows, and $\catA_3$ = the object of composable triples of arrows.
That is a useful point of view, and it sets the meaning in internal category theory for $\catA_0$.

On the other hand, in (enriched category theory), there is a long tradition, starting with the Eilenberg-Kelly paper “Closed Categories” and continuing through Kelly’s basic text, Basic Concepts of Enriched Category Theory, where (the $()_0$ subscript) has a different meaning when applied to (an enriched category).
If $\catA$ is (an enriched category), $\catA_0$ is NOT ($\catA$’s set of objects) (Kelly denotes that by $\ob\catA$, while some other authors use $|\catA|$),
but rather ($\catA$’s underlying ordinary ($\Set$-enriched) category), as defined in section 1.3 of [BCECT].
We wish to emphasize (the enriched categorical point of view), so we adopt its notation.
Thus henceforth $\boxed{\bftwo_0}$ is not the mere set $\{\bot,\top\}$, but rather an isomorph of the ordinary category, and linear order, $\cattwo$.

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$\bftwo$ and $\Set$ as $\text{ccccc}$s - complete and cocomplete cartesian closed categories

We assume as already known the fact that $\bftwo$ and $\Set$ are complete and cocomplete cartesian closed categories, whose operations include:

$\bftwo$ and $\Set$ as $\text{ccccc}$s (complete and cocomplete cartesian closed categories)

		completeness				cocompleteness				internal hom
arity:		nullary	binary	arbitrary		nullary	binary	arbitrary		binary
$\bftwo$		$\top$	$\wedge$	$\forall$ or $\bigwedge$		$\bot$	$\vee$	$\exists$ or $\bigvee$		$\Rightarrow$
$\Set$		$1$	$\times$	$\displaystyle\prod$		$\emptyset$	$+$	$\sum$ or $\displaystyle\coprod$		$[,]$

We denote the internal hom in $\Set$ (the set of functions from, say, set $\setX$ to set $\setY$), often denoted by exponentiation $\setY^\setX$, by $[\setX,\setY]$.
The other operations necessary for completeness and cocompleteness in $\Set$ are equalizers and coequalizers.
For $\Set$ equalizers are easy, while coequalizers involve completing a parallel pair of functions into an equivalence relation, then taking the quotient of that relation.

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References

CWM, Mac Lane, Saunders, Categories for the Working Mathematician (1971,1998)
BCECT, Kelly, Max, Basic Concepts of Enriched Category Theory (1982/2005)
CC, Eilenberg and Kelly, “Closed Categories” (1966)
NAMC, Janelidze and Kelly, “Note on Actions of a Monoidal Category” (2001)

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Draft stuff:

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Relations between the categories

This section is a rough draft:

From (the total order $\cattwo$) to (the $\bftwo$-category $\bftwo$):
The total order $(\cattwo,{\le})$ has
binary products = binary meets = conjunction = $\land$,
and, for each $\objB\in\cattwo$, the order-preserving function $\boxed{-\land\objB : \cattwo\to\cattwo}$
has a right adjoint, denoted $\boxed{\objB\Rightarrow -}$ or $\boxed{[\objB,-]}$ or, combining the previous two notations, $\boxed{[\objB\Rightarrow-]}$ , thus we have:
\[ \pi: \big( \objA\land\objB \le \objC \big) \text{ iff } \big( \objA \le [\objB\Rightarrow\objC] \big), \mkern6em \text{a special case of} \mkern6em \hom {\objectA\tensor\objectB} {(\cal V_0)} {\objectC} \mathop\cong\limits^{\textstyle\pi}_{\text{II, $\S$3}} \hom \objectA {(\cal V_0)} {[\objectB,\objectC]} \mkern6em \text{for $\calV_0 = \cattwo$} \;. \]

From (the $\bftwo$-category $\bftwo$) to (its underlying total order $\bftwo_0$):
Given the internal hom $[,]$, i.e., $\Rightarrow$, define \[ v: \big( \objA\le\objB \big) \text{ iff } \big( \top = [\objA\Rightarrow\objB] \big), \mkern6em \text{which is almost a special case of} \mkern6em \hom \objectA {(\cal V_0)} {\objectB} \mathop\cong\limits^{\textstyle v}_{\text{II (3.12)}} \hom \objectI {(\cal V_0)} {[\objectA,\objectB]} \;. \] (The general cases indicated above) are standard situations in (the theory of closed categories; see section II.3 of [CC]).
The references below the $\cong$ symbols are references to that paper.
The “almost” qualifier is required because (the situation in which it appears) is (a definition of $\le$ in $\cal V_0$), not (an isomorphism between already existing objects).
(The order relation thus defined on $\{\bot,\top\}$) is exactly the same as (that given by fiat on $\{\bot,\top\}$ in the definition of $\cattwo$ as a metacategory, etc.).
Thus $\bftwo_0$ and $\cattwo$ are isomorphic total orders.

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\[\begin{array}{} \boxed{\begin{array}{} \textstyle \ar\cattwo \atop {\Rule{2em}{1px}{1px}} \\ 1_\bot \\ \lt \\ 1_\top \\ \end{array}} & \cong & \boxed{\begin{array}{} {\textstyle \le_2} \atop {\Rule{3em}{1px}{1px}} \\ \langle\bot,\bot\rangle \\ \langle\bot,\top\rangle \\ \langle\top,\top\rangle \\ \end{array}} & \cong & \mkern2em \boxed{\begin{array}{} \textstyle \ar(\bftwo_0) \atop {\Rule{4em}{1px}{1px}} \\ \top=\hom \bot\bftwo\bot \\ \top=\hom \bot\bftwo\top \\ \top=\hom \top\bftwo\top \\ \end{array}} & \to & 1 \\ & \searrow & \big\downarrow & \swarrow && \lrcorner & \big\downarrow \rlap\top \\ 1 & \xrightarrow{\textstyle \langle\objectA,\objectB\rangle} & \boxed{\begin{array}{} \{\bot,\top\} \times \{\bot,\top\} \\ \langle\bot,\bot\rangle \\ \langle\bot,\top\rangle \\ \langle\top,\top\rangle \\ \langle\top,\bot\rangle \\ \end{array}} & {}\rlap{\xrightarrow[\boxed{\begin{array}{} \textstyle \hom - \bftwo - \\ [,]_2 \\ \Rightarrow_2 \\ \le_2 \\ \end{array}} ]{\mkern13em}} &&& \boxed{\begin{array}{} \{\bot,\top\} \\ \bot \\ \top \\ \end{array}} \\ && \llap{\text{projections from cartesian product}} \bigg\downarrow \bigg\downarrow \\ && \boxed{\begin{array}{} \{\bot,\top\} \\ \bot \\ \top \\ \end{array}} \\ \end{array}\]