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\begin{fcdefn}[Functor]
  % Definition 1.4
  \glsnoundefn{func}{functor}{functors}%
  \glsadjdefn{funcal}{functorial}{NA}%
  \glssymboldefn{Cat}%
  Let $\mathcal{C}$ and $\mathcal{D}$ be
  \glspl{cat}. A \emph{functor} $F : \mathcal{C}
  \to \mathcal{D}$ consists of mappings $F : \ob
  \mathcal{C} \to \ob \mathcal{D}$ and $F + \mor
  \mathcal{C} \to \mor \mathcal{D}$ such that:
  \begin{citems}
    \item $F(\dom f) = \dom Ff$
    \item $F(\cod f) = \cod Ff$
    \item $F(\identity A) = \identity{FA}$
    \item $F(fg) = (Ff)(Fg)$ whenever $fg$ is
      defined.
  \end{citems}
  We write $\textbf{Cat}$ for the \gls{cat} of
  \gls{small} \glspl{cat} and the functors between
  them.
\end{fcdefn}

\begin{example}
  \label{eg:1.5}
  % Example 1.5
  \phantom{}
  \begin{enumerate}[(a)]
    \item We have \emph{forgetful} functors $\Gp
      \to \Set$, $\Rng \to \Set$, $\Top \to \Set$,
      \ldots{} or slightly more interestingly,
      $\Rng \to \mathbf{AbGp}$, $\Met \to \Top$,
      $\mathbf{TopGp} \to \Top$, $\mathbf{TopGp}
      \to \Gp$, \ldots
    \item The construction of free groups is a
      \gls{func} $\Set \to \Gp$: given a set $A$,
      $FA$ is the group freely generated by $A$,
      such that every mapping $A \to G$ where
      $G$ has a group structure extends uniquely
      to a homomorphism $FA \to G$. Given $A
      \stackrel{f}{\to} B$, we define $Ff : FA \to
      FB$ to be the unique homomorphism extending
      $A \stackrel{f}{\to} B \injto FB$. If we
      also have $B \stackrel{g}{\to} C$, $F(gf)$
      and $(Fg)(Ff)$ are both homomorphisms
      extending $A \stackrel{f}{\to} B
      \stackrel{g}{\to} C \injto FC$.
    \item Given a set $A$, we define $PA$ to be
      the set of subsets of $A$. Given $f : A \to
      B$, we define $Pf : PA \to PB$ by $Pf(A') =
      {f(a) \st a \in A'} \subseteq B$. So $P$ is
      a \gls{func} $\Set \to \Set$.
    \item
      \glsadjdefn{cont}{contravariant}{\gls{func}}%
      But we also have a \gls{func} $P^* :
      \Set^\op \to \Set$ (or $\Set \to \Set^\op$):
      $P^* A = PA$ and, for $A \stackrel{f}{\to}
      B$, $G^* f : PB \to PA$ is given by $P^* f
      (B') = {a in A \st f(a) \in B'}$. We use the
      term ``contravariant \gls{func}
      $\mathcal{C} \to \mathcal{D}$'' for a
      \gls{func} $\mathcal{C} \to \mathcal{D}
      ^\op$.
    \item Given a vector space $V$ over $k$, we
      write $V^*$ for the space of linear maps $V
      \to k$. Given $f : V \to W$, we write $f^* :
      W^* \to V^*$ for the mapping $\theta \mapsto
      \theta f$. This defines a \gls{func}
      $(\bullet)^* : \Vect_k^\op \to \Vect_k$.
    \item The mapping $\mathcal{C} \mapsto
      \mathcal{C}^\op$, $F \mapsto F$ defines a
      \gls{func} $\Cat \to \Cat$.
    \item A \gls{func} between monoids is a monoid
      homomorphism; a \gls{func} between posets is a
      monotone map.
    \item Given a group $G$, a \gls{func} $G \to
      \Set$ is given by a set $A$ equipped with a
      $G$-action $(g, a) \mapsto g \cdot a$, i.e.
      a permutation representation of $G$.
      Similarly, a \gls{func} $G \to \Vect_k$ is a
      $k$-linear representation of $G$.
    \item The fundamental group construction is a
      \gls{func} $\Pi_1 : \Top_* \to \Gp$, where
      $\Top_*$ is the \gls{cat} of topological
      spaces with basepoints, and morphisms being
      the continuous maps which preserve the
      basepoints.
  \end{enumerate}
\end{example}

\begin{fcdefn}[Natural transformation]
  % Definition 1.6
  \glsnoundefn{natt}{natural
  transformation}{natural transformations}%
  \glssymboldefn{funccat}%
  \glsadjdefn{naty}{naturality}{\gls{func}}%
  \glsnoundefn{natsq}{naturality
  square}{naturality squares}%
  \glsadjdefn{natal}{natural}{transformation}%
  Given \glspl{cat} $\mathcal{C}$ and
  $\mathcal{D}$, and two \glspl{func} $\mathcal{C}
  \twofuncs{F}{G} \mathcal{D}$, a \emph{natural
  transformation} $\alpha : F \to G$ assigns to
  each $A \in \ob \mathcal{C}$ a morphism
  $\alpha_A : FA \to GA$ in $\mathcal{D}$, such
  that for any $A \stackrel{f}{\to} B$ in
  $\mathcal{C}$, the square
  \begin{picmath}
    \begin{tikzcd}
      FA \ar[r, "Ff"] \ar[d, "\alpha_A"]
      &FB \ar[d, "\alpha_B"] \\
      GA \ar[r, "Gf"]
      &GB
    \end{tikzcd}
  \end{picmath}
  commutes (we call this square the
  \emph{naturality square} for $\alpha$ at $f$). Given
  $\alpha$ as above, and $\beta : G \to H$, we
  define $\beta \alpha : F \to H$ by $(\beta
  \alpha)_A = \beta_A \alpha_A$. We write $[
  \mathcal{C}, \mathcal{D}]$ for the \gls{cat} of
  functors $\mathcal{C} \to \mathcal{D}$ and
  natural transformations between them.
\end{fcdefn}

\begin{example}
  \label{eg:1.7}
  % Example 1.7
  \phantom{}
  \begin{enumerate}[(a)]
    \item Given a vector space $V$, we have a
      linear map $\alpha_V : V \to V^{**}$ sending
      $v \in V$ to the linear form $\theta \mapsto
      \theta(v)$ on $V^{**}$. These maps define a
      natural transformation $\identity{\Vect_k}
      \to (\bullet)^{**}$.
    \item There is a natural transformation
      $\alpha : \identity{\Set} \to UF$, where
      $F$ is the free group functor and $U$ is the
      forgetful functor $\Gp \to \Set$, whose
      value at $A$ is the inclusion $A \injto
      UFA$. The naturality square
      \begin{picmath}
        \begin{tikzcd}
          A \ar[r, "f"] \ar[d, "\alpha_A"]
          &B  \ar[d, "\alpha_B"] \\
          UFA \ar[r, "UFf"]
          &UFB
        \end{tikzcd}
      \end{picmath}
      commutes by the definition of $Ff$.
    \item For any $A$, we have a mapping $\eta_A :
      A \to PA$ given by $A\eta_A(a) = \{a\}$.
      This is a natural transformation
      $\identity{\Set} \to P$ since $Pf(\{a\}) =
      \{f(a)\}$ for any $a \in A$.
    \item Given order-preserving maps $P
      \twofuncs{f}{g} Q$ between posets, there
      exists a unique natural transformation $f
      \to g$ if and only if $f(p) \le g(p)$ for
      all $P \in P$.
    \item Given two group homomorphisms $G
      \twofuncs{u}{v} H$, a natural transformation
      $u \to v$ is given by $h \in H$ such that
      $hu(g) = v(g) h$ for all $g \in G$, or
      equivalently $u(g) = h^{-1} v(g) h$, i.e.
      $u$ and $v$ are conjugate homomorphisms. In
      particular, the group of natural
      transformations $u \to u$ is the
      \emph{centraliser} of the image of $u$.
    \item If $A$ and $B$ are $G$-sets considered
      as functors $G \to \Set$, a natural
      transformation $f : A \to B$ is a
      $G$-invariant map, i.e. $f : A \to B$ such
      that $g f(a) = f(g a)$ for all $a \in A$, $g
      \in G$.
