We show various relations between the concepts mentioned in the title.
We begin by recalling one of the definitions of colimit.
If $F : \calJ \to \calC$ is a functor, then its colimit, if it exists,
is a left extension diagram in the 2-category $\Cat$:
\[ \begin{array}{} && \llap{ (\text{ unit category} = {}) \kern.5em } {\mathcal I} \\ & \llap ! \nearrow & \leftadj{ \Big \Uparrow \rlap \iota } & \leftadj\searrow \rlap{\leftadj{\text{colimit }} F} \\ \mathcal J & {} \rlap{ \kern-1em \xrightarrow[\textstyle F]{\kern7em} } &&& \mathcal C \\ \end{array} \]
Now specialize to ($\mathcal J = \mathbf 0$, the empty category),
and ($\boxed{ F = \bigcirc : \mathbf 0 \to \calC }$ the unique (empty) functor).
Then (the above left extension diagram for $(\text{colimit }F)$) specializes to
\[ \begin{array}{} && \llap{ (\text{ unit category} = {}) \kern.5em } {(\mathcal I = \mathbf 1)} \\ & \llap ! \nearrow & \leftadj{ \Big \Uparrow \rlap \iota } & \leftadj\searrow \rlap{ \leftadj{\text{colimit }} \bigcirc } \\ \llap{ (\text{empty category } = {} ) \kern.5em } {\mathbf 0} & {} \rlap{ \kern-1em \xrightarrow[\textstyle \bigcirc]{\kern11em} } &&& \mathcal C \\ \end{array} \]
There is only one possible natural transformation out of (the empty functor $\bigcirc$), thus we have the bijection (*) in:
\[ \boxed{ \begin{array}{} \kern7.5em & \hom \bigcirc {[\mathbf 0, \calC]} {!c} & \buildrel \text{(*)} \over \cong & \mathbf 1 & \kern12em \\ & \llap{\text{definition (colimit $\bigcirc$)}} {\wr\Vert} && {\wr\Vert} \rlap{\text{ definition (initial object = $\bot$)}} \\ & {} \rlap{ \kern-3.8em \hom {\text{(colimit $\bigcirc$)}} \calC c } && \hom \bot \calC c \\ \end{array} } \]
Since this is true (for all $c \in \calC$), we have $\boxed{ \big( \text{colimit } (\bigcirc : \mathbf 0 \to \calC) \big) \cong \bot }$, i.e.,
(an initial object $\bot$) is (a colimit of (the empty functor)).
Note that the above proof only needed the bijection (*) and the definitions of colimit and initial object.
This generalizes the order-theoretic result that, in a preorder,
(a least upper bound, i.e. supreum, for the empty subset) is (a bottom).
For an example of non-uniqueness,
consider a set with two or more elements with the indiscrete (chaotic) preorder, which is certainly not antisymmetric, thus is a preorder but not a partial order.
For such a preorder, every element is both a lub($\emptyset$) and a bottom.
As to the existence of <i>minimal</i> elements, <a href="https://en.wikipedia.org/wiki/Greatest_element_and_least_element">Wikipedia</a> gives two definitions, one for preorders and one for partial orders.
Per the preorder definition, which is the appropriate definition here,
<i>EVERY</i> element is minimal.
Per the (more familiar) definition for partial orders (which makes sense even for preorders, even if it is not the proper definition in those cases),
<i>NO</i> element is minimal.
<hr />
In addition to (initial objects) being (colimits of the smallest possible functor into $\mathcal C$, $\bigcirc : \mathbf 0 \to \mathcal C$),
(initial objects) are also (limits of the largest possible functor into $\mathcal C$, the identity functor $1_{\mathcal C} : \mathcal C \to \mathcal C$),
generalizing the fact that in pre-orders, bottoms are infima, i.e. greatest lower bounds, for the entire pre-order.
(A bottom $\leftadj\bot$ in $\mathcal C$) has (a unique arrow $\boxed{ \bigcirc_c : \leftadj\bot \to c }$) into (each object $c \in \calC$).
(Note that we use the same symbol, $\bigcirc$, both
externally, in $\CAT$, to denote (the unique arrow (a functor)) from (the initial object $\mathbf 0$) to (an arbitrary object, a category, $\mathcal C$) in $\CAT$, and
internally, in $\calC$, to denote (the unique arrow) from (the initial object $\bot$) to (an arbitrary object $c$) in (a given category $\mathcal C$).)
These provide (the projection arrows) necessary to make $\leftadj\bot$ (a limit for the functor $1_{\calC}$).
Two key facts about $\bigcirc$ follow from (the uniqueness condition) in (the definition of initiality): since ($\leftadj\bot$ is initial),
$\bullet$ There is one and only one endoarrow (the unique self-map) $\leftadj\bot \to \leftadj\bot$,
thus $\boxed{ ( \bigcirc_{\leftadj\bot} = 1_{\leftadj\bot} ) : \leftadj\bot \to \leftadj\bot }$.
$\bullet$ The family of arrows $\{\bigcirc_c : \leftadj\bot \to c\}_{c \in \calC}$ collectively form a cone $\boxed{ \bigcirc : {\leftadj\bot}\Delta \Rightarrow 1_{\calC} }$;
the transformation $\bigcirc$ is natural since $\bigcirc_c \gamma = \bigcirc_{c'}$ for any $\gamma : c \to {c'}$ in $\calC$.
\[ \begin{array}{} && \leftadj\bot \\ & \llap{\bigcirc_c} \swarrow && \searrow \rlap{\bigcirc_{c'}} \\ c & {}\rlap{ \kern-.5em \xrightarrow[\textstyle \gamma]{\kern7em} } &&& {c'} \end{array} \]
Thus the family of arrows $\{\bigcirc_c : \leftadj\bot \to c\}_{c \in \calC}$
is closed under post-composition, i.e., is a one-sided ideal.
<hr />
To show ($\boxed{ {\leftadj\bot} [\bigcirc] = <\leftadj\bot, {\leftadj\bot}\Delta \buildrel \bigcirc \over \Rightarrow 1_{\calC}>} $ is a limit for $1_{\calC}$),
we must show (${\leftadj\bot} [\bigcirc]$ is terminal in $(\Delta \downarrow \ulcorner 1_\calC \urcorner)$, the category of cones to $1_\calC$),
i.e. the comma category arising from the displayed cospan in $\CAT$:
\[\begin{array}{cc|cccccccc|c|cccc} &&&& && \boxed{ (\Delta \downarrow \ulcorner 1_\calC \urcorner) } \rlap{ \text{ ( = cones to $1_\calC$) } } \\ &&&& & \llap b \swarrow & & \searrow \llap ! \\ &&&& \calC && \buildrel \textstyle \tau \over \Rightarrow && \mathcal I \\ &&&& & \llap{\Delta} \searrow & \CAT & \swarrow \rlap{ \ulcorner 1_\calC \urcorner } \\ &&&& && [\calC, \calC] \\ \\ \hline &&&& &&&&&& \kern6em & {\leftcat b} \\ {\leftcat b}[\tau] &&&& & {\leftcat b} \Delta & \xrightarrow[\kern2em]{\textstyle \tau} & 1_\calC & & & & & \searrow \rlap{\tau_c} & \rlap{\tau_\gamma} & \searrow \rlap{\tau_{c'}} \\ \llap{ \text{(the generic arrow in $(\Delta \downarrow 1_\calC)$)} \kern2em \leftcat\beta} \Bigg\downarrow &&&& & \llap{\leftcat\beta \Delta} \Bigg \downarrow & [\calC, \calC] & \Bigg\Vert & & & & \llap{\leftcat \beta} \Bigg \downarrow & \calC & c & \xrightarrow[\kern3em]{\gamma} & c' \\ {\leftcat{b'}}[\tau'] &&&& & {\leftcat {b'}} \Delta & \xrightarrow[\textstyle \tau']{\kern2em} & 1_\calC & & & & & \nearrow \rlap{{\tau'}_c} & \rlap{{\tau'}_\gamma} & \nearrow \rlap{{\tau'}_{c'}} \\ &&&& &&&&&& & {\leftcat b'} \\ \\ \hline &&&& &&&&&& \kern6em & {\leftadj\bot} \\ {\leftadj\bot}[\bigcirc] &&&& & {\leftadj\bot} \Delta & \xrightarrow[\kern2em]{\textstyle \bigcirc} & 1_\calC & & & & & \searrow \rlap{(\bigcirc_{\leftadj\bot} {=} 1_{\leftadj\bot})} & & \kern2em \searrow \rlap{\bigcirc_{c}} \\ \llap{ \text{(a special case)} \kern2em \leftcat\beta} \Bigg\downarrow &&&& & \llap{\leftcat\beta \Delta} \Bigg \downarrow & [\calC, \calC] & \Bigg\Vert & & & & \llap{\leftcat \beta} \Bigg \downarrow & \calC & \leftadj\bot & \xrightarrow[\kern3em]{\gamma} & c \\ {\leftcat{b}}[\tau] &&&& & {\leftcat {b}} \Delta & \xrightarrow[\textstyle \tau]{\kern2em} & 1_\calC & & & & & \nearrow \rlap{{\tau}_{\leftadj\bot}} & {} \rlap{\tau_\gamma} & \nearrow \rlap{{\tau}_{c}} \\ &&&& &&&&&& & {\leftcat b} \end{array}\]
Let $\boxed{ \leftcat b[\tau] = \leftcat b[ \leftcat b\Delta \buildrel \textstyle \tau \over \Rightarrow 1_{\calC} ] = <\leftcat b, \leftcat b\Delta \buildrel \textstyle \tau \over \Rightarrow 1_{\calC}> }$ be an arbitrary cone in $(\Delta \downarrow \ulcorner 1_\calC \urcorner)$.
Since ($\tau$ is a cone), we have, for each $c \in \calC$,
\[ \boxed{ \begin{array}{ccccc|c|ccccc} && \leftcat b && && \leftcat b & {} \rlap{ \kern-1em \xrightarrow[\kern9em]{\textstyle \tau_{\leftadj\bot}} } &&& {\leftadj\bot} \\ & \llap{\tau_{\leftadj\bot}} \swarrow & \tau_{\bigcirc_c} & \searrow \rlap{\tau_c} && \text{i.e., reflecting,} & & \llap{\tau_c} \searrow & \tau_{\bigcirc_c} & \swarrow \rlap{ \bigcirc_c } \\ \leftadj\bot & {} \rlap{\kern-1em \xrightarrow[\textstyle \bigcirc_c]{\kern9em}} &&& c & \kern6em & && c \end{array} } \]
Thus $\boxed{ \tau_{\leftadj\bot} : \leftcat b \to {\leftadj\bot} }$ is an arrow in the comma category $(\Delta \downarrow \ulcorner 1_\calC \urcorner)$, $\tau_{\leftadj\bot}: \leftcat b[\tau] \to {\leftadj\bot}[\bigcirc]$.
It remains to show it is the unique such arrow.
Suppose $\boxed{ g : \leftcat b \to {\leftadj\bot} }$ is another such arrow in $(\Delta \downarrow \ulcorner 1_\calC \urcorner)$, $g : \leftcat b[\tau] \to {\leftadj\bot}[\bigcirc]$.
Then we have, for each $c \in {\calC}$ the commutative triangle above the single horizontal line,
while specializing $c$ to be $\leftadj\bot$ gives the triangle below it:
\[ \boxed{ \begin{array}{l|ccccc|c|cc} &&& {} \rlap{ \kern-4em \text{showing} } &&&&& & {} \rlap{ \kern-4em \text{showing} } \\ &&& {} \rlap{ \kern-4em \text{(an initial ${\leftadj\bot} [\bigcirc]$ in $\calC$)} } &&&&&& {} \rlap{ \kern-9em \text{(a limit $\rightadj{ \lim[\pi : {\lim}\leftadj\Delta \Rightarrow \rightcat{1_\calC}] } \in \rightcat{ (\leftadj\Delta \downarrow \ulcorner 1_\calC \urcorner) }$ of $\rightcat{1_\calC}$)} } \\ &&& {} \rlap{ \kern-4em \text{is (a limit of $\rightcat{1_\calC}$)} } &&&&& & {} \rlap{ \kern-4em \text{is (initial in $\calC$)} } \\ \hline \\ \hline \text{existence} & \leftcat b & {} \leftcat{ \rlap{ \kern-1em \xrightarrow [\kern11em]{ \textstyle \boxed{ \exists \; {\rightcat\tau}_{\leftadj\bot} } } } } &&& \leftadj\bot & \kern6em & && \\ && \rightcat{ \llap{\tau_c} \searrow } & {\rightcat\tau}_{\bigcirc_{\rightcat c}} & \swarrow \rlap{ \bigcirc_{\rightcat c} } && & \rightadj\lim & {} \rlap{ \kern-1em \rightadj{ \xrightarrow [\kern9em] { \textstyle \smash{ \boxed{\exists \; \pi_{\rightcat c}} } } } } &&& \rightcat{c \, \forall} \\ &&& \rightcat c && && \\ \hline \text{uniqueness} \\ \hline \text{suppose} & \leftcat b[\tau] & {} \rlap{ \kern-1.5em \xrightarrow[\textstyle \kern4em g \kern4em]{\textstyle (\Delta \downarrow 1_\calC)} } &&& {\leftadj\bot}[\bigcirc] &&&&& \\ \hline \\ \hline & \leftcat b & {}\rlap{ \kern-1em \xrightarrow [\kern11em]{\textstyle g} } &&& \leftadj\bot & \kern6em & && \rightadj\lim \\ \text{then} && \llap{\tau_{\rightcat c}} \searrow & \rightcat{ \forall c }& \swarrow \rlap{ \bigcirc_{\rightcat c} } && && \rightadj{ \llap{\pi_\lim} \swarrow } & \rightadj{ \pi_{ \pi_{\rightcat c} } } & \rightadj{ \searrow \rlap{\pi_{\rightcat c}} } \\ &&& \rightcat c && && \rightadj\lim & {} \rlap{ \kern-1em \rightadj{ \xrightarrow[\textstyle \pi_{\rightcat c}]{\kern9em} } } &&& \rightcat{c \, \forall} \\ \hline \text{thus} & \leftcat b & {} \rlap{ \kern-1em \xrightarrow[\kern11em]{\textstyle g} } &&& \leftadj\bot & & && \rightadj\lim \\ \text{in} && \llap{\tau_{\leftadj\bot}} \searrow & \rightcat{ c = \leftadj\bot } & \swarrow \rlap{ \bigcirc_{\leftadj\bot} = 1_{\leftadj\bot} } && && \rightadj{ \llap{ 1_{\rightadj\lim} = \pi_\lim } \swarrow } & {\rightadj\pi}_f & \rightadj{ \searrow \rlap{ \pi_{\rightcat c} } } \\ \text{particular} &&& \leftadj\bot && && \rightadj\lim & {} \rlap{ \kern-1em \xrightarrow[\textstyle f \, \forall]{\kern9em} } &&& \rightcat{c \, \forall} \end{array} } \]
But $\bigcirc_{\leftadj\bot} = 1_{\leftadj\bot}$.
Thus $g = \tau_{\leftadj\bot}$.
Thus (${\leftadj\bot} [\bigcirc]$ is terminal in $(\Delta \downarrow \ulcorner 1_\calC \urcorner)$,
and thereby a limit of $1_{\calC}$. QED.
<hr />!
The following is the somewhat complicated version of the above used in the Adjoint Functor Theorem to prove that a limit in a certain comma category is initial, and thus constitutes a Left Adjoint.
\[ \boxed{ \begin{array}{l|ccccc|c|cc} &&& {} \rlap{ \kern-4em \text{showing} } &&&&& & {} \rlap{ \kern-5em \text{showing} } \\ &&& {} \rlap{ \kern-4em \text{(an initial ${\leftadj\bot} [\bigcirc]$ in $\calC$)} } &&&&&& {} \rlap{ \kern-14em \text{(a limit $\rightadj{ \lim[\pi : {\lim}\leftadj\Delta \Rightarrow \rightcat{1_{ \leftcat{(l \downarrow \rightadj R)} }}] } \in \rightcat{ (\leftadj\Delta \downarrow \ulcorner 1_{ \leftcat{(l \downarrow \rightadj R)} } \urcorner) }$ of $1_{ \leftcat{(l \downarrow \rightadj R)} }$)} } \\ &&& {} \rlap{ \kern-4em \text{is (a limit of $\rightcat{1_\calC}$)} } &&&&& & {} \rlap{ \kern-7em \text{is (initial in $\leftcat{(l \downarrow \rightadj R)}$)} } \\ \hline \\ \hline \text{existence} & \leftcat b & {} \leftcat{ \rlap{ \kern-1em \xrightarrow [\kern11em]{ \textstyle \boxed{ \exists \; {\rightcat\tau}_{\leftadj\bot} } } } } &&& \leftadj\bot & \kern6em & && \\ && \rightcat{ \llap{\tau_c} \searrow } & {\rightcat\tau}_{\bigcirc_{\rightcat c}} & \swarrow \rlap{ \bigcirc_{\rightcat c} } && & { \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] } } & {} \rlap{ \kern-1em \rightadj{ \xrightarrow [\kern18em] { \textstyle \smash{ \boxed{ \exists \; \pi_{ \rightcat r \leftcat{[\kappa]} } } } } } } &&& \rightcat{ r \leftcat{[\kappa]} \, \forall } \\ &&& \rightcat c && && \\ \hline \text{uniqueness} \\ \hline \text{suppose} & \leftcat b[\tau] & {} \rlap{ \kern-1.5em \xrightarrow[\textstyle \kern4em g \kern4em]{\textstyle (\Delta \downarrow 1_\calC)} } &&& {\leftadj\bot}[\bigcirc] &&&&& \\ \hline \\ \hline & \leftcat b & {}\rlap{ \kern-1em \xrightarrow [\kern11em]{\textstyle g} } &&& \leftadj\bot & \kern6em & && \rightadj{ \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] } } \\ \text{then} && \llap{\tau_{\rightcat c}} \searrow & \rightcat{ \forall c }& \swarrow \rlap{ \bigcirc_{\rightcat c} } && && \rightadj{ \llap{\pi_{ \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] } } } \swarrow } & \rightadj{ \pi_{ \pi_{ \rightcat r \leftcat{[\kappa]} } } } & \rightadj{ \searrow \rlap{ \pi_{ \rightcat r \leftcat{[\kappa]} } } } \\ &&& \rightcat r \leftcat{[\kappa]} && && \rightadj{ \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] } } & {} \rlap{ \kern-1em \rightadj{ \xrightarrow[\textstyle \pi_{ \rightcat r \leftcat{[\kappa]} }]{\kern18em} } } &&& \rightcat{ r \leftcat{[\kappa]} \, \forall } \\ \hline \text{thus} & \leftcat b & {} \rlap{ \kern-1em \xrightarrow[\kern11em]{\textstyle g} } &&& \leftadj\bot & & && \rightadj{ \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] } } \\ \text{in} && \llap{\tau_{\leftadj\bot}} \searrow & \rightcat{ c = \leftadj\bot } & \swarrow \rlap{ \bigcirc_{\leftadj\bot} = 1_{\leftadj\bot} } && && \rightadj{ \llap{ 1_{ \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] } } = \pi_{ \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] } } } \swarrow } & {\rightadj\pi}_f & \rightadj{ \searrow \rlap{ \pi_{ \rightcat r \leftcat{[\kappa]} } } } \\ \text{particular} &&& \leftadj\bot && && \rightadj{ \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] } } & {} \rlap{ \kern-1em \xrightarrow[\textstyle f \, \forall]{\kern18em} } &&& \rightcat{ r \leftcat{[\kappa]} \, \forall } \\ \end{array} } \]
<hr />
And here is a very general version of the argument,
showing how (the situation) can perspicuously be viewed 2-categorically,
as ( two (horizontal compositions) in (the 2-category $\CAT$) ).
(Writing $\rightadj{\boxed \lim}$ as short for $\rightadj{ \boxed{ \lim \rightcat{1_\calC} } }$.)
\[ \boxed{ .\begin{array} {ccccccccc|c} && \calI &&&& \calI && \\ & \llap{!} \nearrow & \rightadj{ \llap{\pi} \swarrow \rlap{\kern-1.5em \swarrow} } & \rightadj{ \searrow \rlap{{\lim}} } && \llap{!} \nearrow & \rightadj{ \searrow \rlap{\kern-1.5em \searrow \kern0em \pi} } & \rightadj{ \searrow \rlap{{\lim}} } & & \CAT \\ \rightcat{ \calC \rlap{ \kern0em \xrightarrow[\textstyle 1_\calC]{\kern11em} } } &&&& \rightcat{ \calC \rlap{ \kern0em \xrightarrow[\textstyle 1_\calC]{\kern11em} } } &&&& \rightcat\calC \\ \hline &&&& \rightadj\lim \\ &&& .\swarrow && \rightadj{ \searrow \rlap{\lim \pi = \pi_\lim} } && \\ && \rightadj{{\lim}} && {} \rlap{ \kern-2em \rightadj{ \boxed{\pi \pi = \pi_\pi} } } && \rightadj{{\lim}} &&& \rightcat{[ \calC, \calC ]} \\ &&& \rightadj{ \llap{ \pi = \rightcat{1_\calC} \pi } \searrow } && \rightadj{ \swarrow \rlap{ \pi \rightcat{1_\calC} = \pi } } \\ &&&& \rightcat{1_\calC} \\ \hline {} \rlap{ \kern1em \text{Thus, by (the universal property)} } \\ {} \rlap{ \kern1em \text{of (the $\rightadj{ \pi = \pi \rightcat{1_\calC} }$ at the lower right),} } \\ {} \rlap{ \kern4em \boxed{ \rightadj{\pi_\lim = 1_\lim : \lim \to \lim} } \, . } \\ {} \rlap{ \kern-1em \text{The important point here is the confluence of:} } \\ {} \rlap{ \kern2em \text{the self-application (squaring) of} } \\ {} \rlap{ \text{( (the $\rightadj\pi$ for $\rightadj{ \lim(\rightcat{1_\calC}) }$), an endo-2-cell on $\rightcat\calC$ ),} } \\ {} \rlap{ \kern0em \text{and ( (the universal property) of (that $\rightadj\pi$) ).} } \\ \hline && \calI &&&& \calI && \\ & \nearrow & \rightcat{ \llap f \swarrow \rlap{\kern-1.5em \swarrow} } & \rightadj{ \searrow \rlap{{\lim}} } && \llap{!} \nearrow & \rightadj{ \searrow \rlap{\kern-1.5em \searrow \kern0em \pi} } & \rightadj{ \searrow \rlap{{\lim}} } && \CAT \\ \rightcat{ \calI \rlap{ \kern0em \xrightarrow[\textstyle c]{\kern11em} } } &&&& \rightcat{ \calC \rlap{ \kern0em \xrightarrow[\textstyle 1_\calC]{\kern11em} } } &&&& \rightcat\calC \\ \hline &&&& \rightadj\lim \\ &&& \rightcat\swarrow && \rightadj{ \searrow \rlap{\lim \pi = \pi_\lim} } && \\ && \rightadj{{\lim}} && {} \rlap{ \kern-2em \rightcat{ \boxed{f \rightadj\pi = {\rightadj\pi}_f} } } && \rightadj{{\lim}} &&& \rightcat{ [ \calI, \calC ] \cong \calC } \\ &&& \rightadj{ \llap{ \pi_{\rightcat c} = \rightcat c \pi } \searrow } && \rightcat{ \swarrow \rlap{ f \rightcat{1_\calC} = f } } \\ &&&& \rightcat c \\ \end{array} } \]
Of course in each case what counts is
the naturality of the right-hand occurrence of $\rightadj\pi$.
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