$\quad$
\begin{enumerate}
\item There is an endofunctor $P \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ that sends a set $A$ to its power set $PA = { A’ \subset A}$ and a function $f \colon A \to B$ to the direct-image function $f_* \colon PA \to PB$ that sends $A’ \subset A$ to $f(A’) \subset B$.
\item Each of the categories listed in Example \ref{exs:concrete-categories} has a forgetful functor, a general term that is used for any functor that forgets structure, whose codomain is the category of sets. For example, $U \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ sends a group to its underlying set and a group homomorphism to its underlying function. The functor $U \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ sends a space to its set of points. There are two natural forgetful functors $V,E \colon \textup{\textsf{cat}} \rightrightarrows \textup{\textsf{cat}}$ that send a graph to its vertex or edge sets, respectively; if desired, these can be combined to define a single functor $V \sqcup E \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ that carries a graph to the disjoint union of its vertex and edge sets. These mappings are functorial because in each instance a morphism in the domain category has an underlying function.
\item There are intermediate forgetful functors $\textup{\textsf{cat}}R \to \textup{\textsf{cat}}$ and $\textup{\textsf{cat}} \to \textup{\textsf{cat}}$ that forget some but not all of the algebraic structure. The inclusion functors $\textup{\textsf{cat}} \hookrightarrow \textup{\textsf{cat}}$ and $\textup{\textsf{cat}} \hookrightarrow \textup{\textsf{cat}}$ may also be regarded as forgetful.'' Note that the latter two, but neither of the former, are injective on objects: a group is either abelian or not, but an abelian group might admit the structure of a ring in multiple ways.
\item Similarly, there are forgetful functors $\textup{\textsf{cat}} \to \textup{\textsf{cat}}_*$ and $\textup{\textsf{cat}} \to \textup{\textsf{cat}}_*$ that take the basepoint to be the identity and zero elements, respectively. These assignments are functorial because group and ring homomorphisms necessarily preserve these elements.
\item There are functors $\textup{\textsf{cat}} \to \textup{\textsf{cat}}$ and $\textup{\textsf{cat}}_* \to \textup{\textsf{cat}}_*$ that act as the identity on objects and send a (based) continuous function to its homotopy class.
\item The \emph{fundamental group} defines a functor $\pi_1 \colon \textup{\textsf{cat}}_* \to \textup{\textsf{cat}}$; a continuous function $f\colon (X,x) \to (Y,y)$ of based spaces induces a group homomorphism $f_* \colon \pi_1(X,x) \to \pi_1(Y,y)$ and this assignment is functorial, satisfying the two functoriality axioms described above. A precise expression of the statement thatthe fundamental group is a homotopy invariant’’ is that this functor factors through the functor $\textup{\textsf{cat}}* \to \textup{\textsf{cat}}*$ to define a functor $\pi_1 \colon \textup{\textsf{cat}}* \to \textup{\textsf{cat}}$.
\item A related functor $\Pi_1 \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ assigns an unbased topological space its fundamental groupoid, the category defined in Example \ref{exs:groupoid}\eqref{ex:fundamental}. A continuous function $f \colon X \to Y$ induces a functor $f_* \colon \Pi_1(X) \to \Pi_1(Y)$ that carries a point $x \in X$ to the point $f(x) \in Y$. This mapping extends to morphisms in $\Pi_1(X)$ because continuous functions preserve paths and path homotopy classes.
\item For each $n \in \mathbb{Z}$, there are functors $Z_n, B_n, H_n \colon \textup{\textsf{cat}}R \to \textup{\textsf{cat}}_R$. The functor $Z_n$ computes the $n$-cycles defined by $Z_nC\bullet = \textup{fun}(d \colon C_n \to C_{n-1})$. The functor $B_n$ computes the $n$-boundary defined by $B_nC_\bullet = \textup{fun}(d \colon C_{n+1} \to C_n)$. The functor $H_n$ computes the $n$th homology $H_nC_\bullet \coloneqq Z_nC_\bullet / B_nC_\bullet$. We leave it to the reader to verify that each of these three constructions is functorial. Considering all degrees simultaneously, the cycle, boundary, and homology functors assemble into functors $Z_,B_, H_* \colon \textup{\textsf{cat}}R \to \textup{\textsf{cat}}_R$ from the category of chain complexes to the category of graded $R$-modules. The singular homology of a topological space is defined by precomposing $H$ with a suitable functor $\textup{\textsf{cat}} \to \textup{\textsf{cat}}_R$.
\item There is a functor $F \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ that sends a set $X$ to the free group on $X$. This is the group whose elements are finite words'' whose letters are elements $x \in X$ or their formal inverses $x^{-1}$, modulo an equivalence relation that equates the words $xx^{-1}$ and $x^{-1}x$ with the empty word. Multiplication is by concatenation, with the empty word serving as the identity. This is one instance of a large family offree’’ functors studied in Chapter \ref{ch:adjunction}.
\item The chain rule expresses the functoriality of the derivative. Let $\textup{\textsf{cat}}_$ denote the category whose objects are pointed finite-dimensional Euclidean spaces $(\mathbb{R}^n,a)$—or, better, open subsets thereof—and whose morphisms are pointed differentiable functions. The total derivative of $f \colon \mathbb{R}^n \to \mathbb{R}^m$, evaluated at the designated basepoint $a \in \mathbb{R}^n$, gives rise to a matrix called the Jacobian matrix defining the directional derivatives of $f$ at the point $a$. If $f$ is given by component functions $f_1,\ldots, f_m \colon \mathbb{R}^n \to \mathbb{R}$, the $(i,j)$-entry of this matrix is $\frac{\partial}{\partial x_j} f_i(a)$. This defines the action on morphisms of a functor $D \colon \textup{\textsf{cat}}* \to \textup{\textsf{cat}}\mathbb{R}$; on objects, $D$ assigns a pointed Euclidean space its dimension. Given $g \colon \mathbb{R}^m \to \mathbb{R}^k$ carrying the designated basepoint $f(a) \in \mathbb{R}^m$ to $gf(a) \in \mathbb{R}^k$, functoriality of $D$ asserts that the product of the Jacobian of $f$ at $a$ with the Jacobian of $g$ at $f(a)$ equals the Jacobian of $gf$ at $a$. This is the chain rule from multivariable calculus.$\textup{\textsf{cat}}*$ to the category of real vector spaces that sends a pointed manifold to its tangent space.}
\item Any commutative monoid $M$ can be used to define a functor $M^{-} \colon \textup{\textsf{cat}}* \to \textup{\textsf{cat}}$. Writing $n_+ \in \textup{\textsf{cat}}*$ for the set with $n$ non-basepoint elements, define $M^{n+}$ to be $M^n$, the $n$-fold cartesian product of the set $M$ with itself. By convention, $M^{0+}$ is a singleton set. For any based map $f \colon m+ \to n_+$, define the $i$th component of the corresponding function $M^f \colon M^m \to M^n$ by projecting from $M^m$ to the coordinates indexed by elements in the fiber $f^{-1}(i)$ and then multiplying these using the commutative monoid structure; if the fiber is empty, the function $M^f$ inserts the unit element in the $i$th coordinate. Note each of the sets $M^n$ itself has a basepoint, the $n$-tuple of unit elements, and each of the maps in the image of the functor are based. It follows that the functor $M^{-}$ lifts along the forgetful functor $U \colon \textup{\textsf{cat}}_* \to \textup{\textsf{cat}}$.