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\begin{lemma}[]
  % Lemma 2.8
  \phantom{}
  \begin{cenum}[(i)]
    \item If $\mathcal{C}$ has \glspl{equaler}
      (i.e. every pair of parallel arrows has an
      \gls{equaler}), then any \gls{deting} family in
      $\mathcal{C}$ is \gls{seping}.
    \item If $\mathcal{C}$ is \gls{bal}, then any
      \gls{seping} family in $\mathcal{C}$ is
      \gls{deting}.
  \end{cenum}
\end{lemma}

\begin{proof}
  \phantom{}
  \begin{enumerate}[(i)]
    \item Suppose $\mathcal{G}$ is a \gls{deting}
      family, and suppose $A \twofuncs fg B$
      satisfy the hypothesis of
      \cref{defn:2.7}(a). Let $E \stackrel{e}{\to}
      A$ of $(f, g)$: then any $G
      \stackrel{h}{\to} A$ with $G \in
      \mathcal{G}$ factors uniquely through $e$,
      so $e$ is an isomorphism, so $f = g$.
    \item Suppose $\mathcal{G}$ is \gls{seping},
      and $A \stackrel{f}{\to} B$ satisfies the
      hypothesis of \cref{defn:2.7}(b). If $C
      \twofuncs gh A$ satisfy $fg = fh$, then any
      $G \stackrel{k}{\to} C$ with $G \in
      \mathcal{G}$ satisfies $gk = hk$, since both
      are factorisations of $fgk$ through $f$. So
      $g = h$; hence $f$ is \gls{monic}.

      Similarly, if $B \twofuncs lm D$ satisfy $lf
      = mf$, then any $G \stackrel{n}{\to} B$
      satisfies $ln = mn$, since it factors
      through $f$, so $l = m$ and hence $f$ is
      epic. Since $\mathcal{C}$ is \gls{bal}, $f$
      is an isomorphism. \qedhere
  \end{enumerate}
\end{proof}

\begin{example}
  \label{eg:2.9}
  % Example 2.9
  \phantom{}
  \begin{enumerate}[(a)]
    \item In $\Set$, $1 = \{*\}$ is a \gls{sep}
      and a \gls{det}, since $\Set(1, \bullet)$ is
      isomorphic to the identity \gls{func}.
      Also, $2 = \{0, 1\}$ is a co\gls{sep} and a
      co\gls{det}, since it represents $P^* :
      \Set^\op \to \Set$.
    \item In $\Gp$ (respectively $\Rng$), $\Zbb$
      (respectively $\Zbb[X]$) is a separator and
      a detector, since it represents the
      forgetful \gls{func}.

      But $\Gp$ has no co\gls{sep} or co\gls{det}
      set: given any set $\mathcal{G}$ of groups,
      there is a simple group $H$ with $\card H >
      \card G$ for all $G \in \mathcal{G}$, so the
      only homomorphisms $H \to G$ with $G \in
      \mathcal{G}$ are trivial.
    \item For any \gls{small} \gls{cat}
      $\mathcal{C}$, the set $\{\mathcal{C}(A,
      \bullet) \st A \in \ob \mathcal{C}\}$ is
      \gls{seping} and \gls{deting} in
      $\funccat[\mathcal{C}, \Set]$. This uses
      \nameref{yoneda} and \cref{lemma:1.8} (for
      the detecting case).
    \item In $\Top$, $1$ is a \gls{sep} since it
      represents $U : \Top \to \Set$. But $\Top$
      has no \gls{deting} set of objects: given a
      set $\mathcal{G}$ of spaces, choose $\kappa
      > \card X$ for all $X \in \mathcal{G}$, and
      let $Y$ and $Z$ be a set of $\card \kappa$.
      Give $Y$ the discrete topology and for $Z$,
      we set the closed sets be $Z$ plus all the
      subsets of $\card \kappa$. The identity $Y
      \to Z$ is continuous, but not a
      homeomorphism, but its restriction to any
      subset of $\card < \kappa$ is a
      homeomorphism, so $\mathcal{G}$ can't
      detect the fact that $f$ isn't an
      isomorphism.
    \item Let $\mathcal{G}$ be the \gls{cat} whose
      objects are the ordinals, with identities
      plus two morphisms $\alpha \twofuncs fg
      \beta$ whenever $\alpha < \beta$ with
      composition defined by $ff = fg = gf = gg =
      f$.
      
      Then $0$ is a \gls{det} for
      $\mathcal{C}$: it can tell that $0
      \twofuncs fg \alpha$ aren't isomorphisms
      since neither factors through the other, and
      if $0 < \alpha < \beta$ it can tell that
      $\alpha \twofuncs fg \beta$ aren't
      isomorphisms since $0 \stackrel{g}{\to}
      \beta$ doesn't factor through either.

      But
      $\mathcal{C}$ has no separating set: if
      $\mathcal{G}$ is any set of ordinals, choose
      $\alpha > \beta$ for all $\beta \in
      \mathcal{G}$ and then $\mathcal{G}$ can't
      separate $\alpha \twofuncs fg \alpha + 1$.
  \end{enumerate}
\end{example}

By definition, the \glspl{func}
$\mathcal{C}(A, \bullet) : \mathcal{C} \to
\Set$ preserve \glspl{mono}, but they don't
always preserve \glspl{epi}.

\begin{fcdefn}[Projective]
  \label{defn:2.10}
  \glsadjdefn{proj}{projective}{object}%
  \glsadjdefn{pinj}{injective}{object}%
  % Definition 2.10
  We say an object $P$ in a \gls{locsm} \gls{cat}
  $\mathcal{C}$is \emph{projective} if
  $\mathcal{C}(P, \bullet)$ preserves \glspl{epi},
  i.e. if given
  \begin{picmath}
    \begin{tikzcd}
      & P \ar[d, "f"] \\
      Q \ar[r, "g", two heads]
      & R
    \end{tikzcd}
  \end{picmath}
  there exists $h : P \to Q$ with $gh = f$.
  Dually, $P$ is \emph{injective} if it's
  projective in $\mathcal{C}^\op$.

  If $P$ satisfies this condition for all $g$ in
  some class $\mathcal{E}$ of \glspl{epi}, we call
  it \emph{$\mathcal{E}$-projective}.
\end{fcdefn}

\glsadjdefn{pws}{pointwise}{NA}%
In $\funccat[\mathcal{C}, \Set]$, we consider the
class of \emph{pointwise epimorphisms}, i.e. those
$\alpha$ such that $\alpha_A$ is surjective for
all $A$.

\begin{fccoro}[]
  % Corollary 2.11
  If $C$ is \gls{small}, then \glspl{func} of the
  form $\mathcal{C}(A, \bullet)$ are \gls{pws}
  \gls{proj} in $\funccat[\mathcal{C}, \Set]$.
\end{fccoro}

\begin{proof}
  Immediate from \nameref{yoneda}; given
  \begin{picmath}
    \begin{tikzcd}
      & \mathcal{C}(A, \bullet) \ar[d, "\alpha"]
      \\
      Q \ar[r, "\beta", two heads]
      & R
    \end{tikzcd}
  \end{picmath}
  with $\beta$ \gls{pws} \gls{epic}, $\Phi(\alpha)
  \in RA$ is $\beta_A(y)$ for some $y \in QA$, so
  $\beta\Psi(y) = \alpha$.
\end{proof}

``$\funccat[\mathcal{C}, \Set]$ has enough
\gls{pws} projectives'':

\begin{fcprop}[]
  Assuming:
  - $\mathcal{C}$ is \gls{small}
  - $F : \mathcal{C} \to \Set$
  Then:
  there exists a \gls{pws} \gls{epi} $P \epic F$
  where $P$ is \gls{pws} \gls{proj}.
\end{fcprop}

\begin{proof}
  Set $P = \coprod_{(A, x)} \mathcal{C}(A,
  \bullet)$ where the disjoint union is over all
  pairs $(A, x)$ with $A \in \ob \mathcal{C}$ and
  $x \in FA$. A morphism $P \to Q$ is uniquely
  determined by a family of morphisms
  $\mathcal{C}(A, \bullet) \to Q$.
. Hence $P$ is \gls{pws} \gls{proj}, since all the
  $\mathcal{C}(A, \bullet)$ are. But we have
  $\alpha : P \to F$ whose $(A, x)$-th component
  is $\Psi(x) : \mathcal{C}(A, \bullet) \to F$ and
  this is \gls{pws} \gls{epic} since any $x \in
  FA$ appears as $\Psi(x)(\identity{A})$.
\end{proof}
