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\begin{visiblefc}{6deace856624}
\begin{proof}[Proof of \nameref{yoneda}(ii)]
  \divcloze{Suppose for the moment that $\mathcal{C}$ is
  \gls{small}, so that $\funccat[\mathcal{C}, \Set]$ is
  \gls{locsm}. Given two \glspl{func} $\mathcal{C}
  \times \funccat[\mathcal{C}, \Set] \to \Set$: the
  first sends an object $(A, F)$ to $FA$, and a
  morphism $(A \stackrel{f}{\to} A', F
  \stackrel{\alpha}{\to} F')$ to the diagonal of
  \begin{picmath}
    \begin{tikzcd}[ampersand replacement=\&]
      FA \ar[r, "Ff"] \ar[d, "\alpha_A"]
      \& FA' \ar[d, "\alpha_{A'}"] \\
      F'A \ar[r, "F'f"]
      \& F'A'
    \end{tikzcd}
  \end{picmath}
  The second is the composite
  \[
    \mathcal{C} \times \funccat[\mathcal{C}, \Set]
    \stackrel{Y \times 1}{\longrightarrow}
    \funccat[\mathcal{C}, \Set]^\op \times
    \funccat[\mathcal{C}, \Set]
    \stackrel{\funccat[\mathcal{C}, \Set](\bullet,
    \bullet)}{\longrightarrow} \Set
  \]
  where $Y$ is a \gls{yonem}. Then $\Phi$ and
  $\Psi$ define a \gls{nati} between these two.

  In elementary terms, this says that if $x \in
  FA$, and $x' \in F'A'$ is its image under the
  diagonal, then $\Psi(x')$ is the composite
  \[
    \mathcal{C}(A', \bullet)
    \stackrel{\mathcal{C}(f,
    \bullet)}{\longrightarrow} \mathcal{C}(A,
    \bullet) \stackrel{\Psi(x)}{\longrightarrow} F
    \stackrel{\alpha}{\longrightarrow} F'
  .\]
  This makes sense without the assumption that
  $\mathcal{C}$ is \gls{small}, and it's true
  since the composite maps
  \[
    \identity{A'} \mapsto f \mapsto (Ff)(x) \mapsto
    \alpha_{A'}(Ff)(x)
  . \qedhere \]
  }
\end{proof}
\end{visiblefc}

\begin{example}
  \label{eg:2.6}
  % Example 2.6
  \phantom{}
  \begin{enumerate}[(a)]
    \item The forgetful \gls{func} $\Gp \to \Set$
      is \gls{reped} by $(\Zbb, 1)$, $\Rng \to
      \Set$ is \gls{reped} by $(\Zbb[X], X)$,
      $\Top \to \Set$ is \gls{reped} by $(\{*\},
      *)$.
    \item The \gls{func} $\mathcal{P}^* : \Set^\op \to \Set$
      is \gls{reped} by $(\{0, 1\}, \{1\})$. This
      is the bijection between subsets of $A$ and
      functions $A \stackrel{f}{\to} \{0, 1\}$,
      and it's natural. But $\mathcal{P} : \Set \to \Set$ is
      not \gls{reple}, since $P(\{*\})$ isn't a
      singleton.
    \item The \gls{func} $\Omega : \Top^\op \to
      \Set$ sending $X$ to the set of open subsets
      of $X$, and $X \stackrel{f}{\to} Y$ to
      $f^{-1} : \Omega(Y) \to \Omega(X)$ is
      \gls{reple} by the \emph{Sierpinski space}
      $\Sigma = \{0, 1\}$ with $\{1\}$ open but
      $\{0\}$ not open. This works since
      continuous maps $X \to \Omega$ are the
      characteristic functions of open subsets of
      $X$.
    \item The \gls{func} $(\bullet)^* : \Vect_k
      \to \Vect_k$ isn't \gls{reple}, but its
      composite with $\Vect_k \to \Set$ is
      \gls{reped} by $k$.
    \item For a group $G$ considered as a
      $1$-object \gls{cat}, the unique \gls{reple}
      \gls{func} $G \to \Set$ is the \emph{Cayley
      representation}: $G$ acting on itself by
      multiplication.
    \item \glsnoundefn{prod}{product}{products}%
      \glsnoundefn{coprod}{coproduct}{coproducts}%
      \glsnoundefn{catprod}{categorical
      coproduct}{categorical coproducts}%
      \glsnoundefn{catcoprod}{categorical
      coproduct}{categorical coproducts}%
      Given two objects $A, B$ in a
      \gls{locsm} \gls{cat} $\mathcal{C}$, we have
      a \gls{func} $\mathcal{C}^\op \to \Set$
      sending $C$ to $\mathcal{C}(C, A) \times
      \mathcal{C}(C, B)$. If this \gls{func} is
      \gls{reple}, we call the representing object
      a \emph{categorical product} $A \times B$
      and write $(\pi_1 : A \times B \to A, \pi_2
      : A \times B \to B)$ for the universal
      element. Its defining property is that given
      any pair $(f : C \to A, g : C \to B)$, there
      is a unique isomorphism $h : C \to A \times
      B$ such that $\pi_q h = f$ and $\pi_2 h =
      g$.

      Dually, we have the notion of
      \emph{coproduct} $A + B$ with coprojections
      $\gamma_1 : A \to A + B$, $\gamma_2 : B \to
      A + B$.
    \item Given a parallel pair $A \twofuncs fg B$
      in a \gls{locsm} \gls{cat} $\mathcal{C}$, we
      have a \gls{func} $F : \mathcal{C}^\op \to
      \Set$ sending $C$ to $\{h : C \to A \st fh =
      gh\}$ and defined on morphisms in the same
      way as $\mathcal{C}(\bullet, A)$.
      
      \glsnoundefn{equaler}{equaliser}{equalisers}%
      \glsnoundefn{coequaler}{coequaliser}{coequalisers}%
      \glsadjdefn{reg}{regular}{morphism}%
      \glsadjdefn{rege}{regular epi}{morphism}%
      A \gls{rep} of this \gls{func} is called an
      \emph{equaliser} of $(f, g)$: it consists of
      $E \stackrel{e}{\to} A$ satisfying $fe = ge$,
      and such that any $h$ with $fh = gh$ factors
      uniquely as $ek$. Note that $e$ is
      \gls{monic}; we call a \gls{mono}
      \emph{regular} if it occurs as an equaliser.

      Dually, we have the notions of
      \emph{coequaliser} and \emph{regular epi}.
  \end{enumerate}
\end{example}

In $\Set$, \glspl{prod} are just cartesian products
(also in $\Gp$, $\Rng$, $\Top$, \ldots).
\Glspl{coprod} in $\Set$ are disjoint unions $A
\amalg B = (A \times \{0\}) \cup (B \times
\{1\})$. In $\Gp$, \glspl{coprod} are free
products $G * H$.

In $\Set$, the \gls{equaler} of $A \twofuncs fg B$
is the inclusion of $\{a \in A \st f(a) = g(a)\}$
and the \gls{coequaler} of $(f, g)$ is the
quotient of $B$ by the smallest equivalence
relation containing $\{(f(a), g(a)) \st a + A\}$.

Note that in $\Set$, all \glspl{mono} and all
\glspl{epi} are \gls{reg}, but in $\Top$, a
\gls{mono} $X \stackrel f\to Y$ is \gls{reg} if
and only if $X$ is topologised as a subspace of
$Y$. An \gls{epi} $X \stackrel f\to Y$ is
\gls{reg} if and only if $Y$ is topologised as a
quotient of $X$.

Note that if $f$ is both \gls{reg} \gls{monic} and
\gls{reg} \gls{epic}, then it's an isomorphism
since the pair $(g, h)$ of which its \gls{equaler}
must satisfy $g = h$.

\begin{warning*}
  The following terminology is not standard. These
  are usually (both!) referred to as ``generating'', but
  to avoid confusion, in this course we will refer
  to them with separate names.
\end{warning*}

\begin{fcdefn}[Separating / generating family]
  \label{defn:2.7}
  % Definition 2.7
  \glsadjdefn{seping}{separating}{family}%
  \glsadjdefn{deting}{detecting}{family}%
  \glsnoundefn{sep}{separator}{separators}%
  \glsnoundefn{det}{detector}{detectors}%
  Let $\mathcal{G}$ be a family of objects of a
  \gls{locsm} \gls{cat} $\mathcal{C}$.
  \begin{cenum}[(a)]
    \item We say $\mathcal{G}$ is a
      \emph{separating family} if the \glspl{func}
      $\mathcal{C}(G, \bullet)$, $G \in
      \mathcal{G}$ are jointly \gls{faith}, i.e.
      given a parallel pair $A \twofuncs fg B$,
      the equations $fh = gh$ for all $h : G \to
      A$ with $G \in \mathcal{G}$ imply $f = g$.
    \item We say $\mathcal{G}$ is a \emph{detecting
      family} if the $\mathcal{G}(G, \bullet)$
      jointly reflect isomorphisms, i.e. given $A
      \stackrel f\to B$, if every $G \stackrel
      g\to B$ with $G \in \mathcal{G}$ factors
      uniquely through $f$, then $f$ is an
      isomorphism.
  \end{cenum}
  If $\mathcal{G} = \{G\}$, we call $G$ a
  \emph{separator} or a \emph{detector}.
\end{fcdefn}