\begin{enumerate}
\item There is an endofunctor that sends a set to its power set and a function to the direct-image function that sends to .
\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, sends a group to its underlying set and a group homomorphism to its underlying function. The functor sends a space to its set of points. There are two natural forgetful functors that send a graph to its vertex or edge sets, respectively; if desired, these can be combined to define a single functor 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 $\textup{\textsf{cat}}R \to \textup{\textsf{cat}}\textup{\textsf{cat}} \to \textup{\textsf{cat}}\textup{\textsf{cat}} \hookrightarrow \textup{\textsf{cat}}\textup{\textsf{cat}} \hookrightarrow \textup{\textsf{cat}}$ 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 that
the fundamental group is a homotopy invariant’’ is that this functor factors through the functor $\textup{\textsf{cat}}* \to \textup{\textsf{cat}}*\pi_1 \colon \textup{\textsf{cat}}* \to \textup{\textsf{cat}}$.
\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 $f \colon X \to Yf_* \colon \Pi_1(X) \to \Pi_1(Y)x \in Xf(x) \in Y\Pi_1(X)$ because continuous functions preserve paths and path homotopy classes.
\item For each $n \in \mathbb{Z}Z_n, B_n, H_n \colon \textup{\textsf{cat}}R \to \textup{\textsf{cat}}_RZ_nn$-cycles defined by $Z_nC\bullet = \textup{fun}(d \colon C_n \to C_{n-1})B_nn$-boundary defined by . The functor computes the th homology . 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 $Z_,B_, H_* \colon \textup{\textsf{cat}}R \to \textup{\textsf{cat}}_RRH\textup{\textsf{cat}} \to \textup{\textsf{cat}}_R$.
\item There is a functor $F \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}X$ to the free group on . 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 of
free’’ functors studied in Chapter \ref{ch:adjunction}.
\item The chain rule expresses the functoriality of the derivative. Let $\textup{\textsf{cat}}_(\mathbb{R}^n,a)$—or, better, open subsets thereof—and whose morphisms are pointed differentiable functions. The total derivative of , evaluated at the designated basepoint , gives rise to a matrix called the Jacobian matrix defining the directional derivatives of at the point . If is given by component functions , the -entry of this matrix is . This defines the action on morphisms of a functor $D \colon \textup{\textsf{cat}}* \to \textup{\textsf{cat}}\mathbb{R}Dg \colon \mathbb{R}^m \to \mathbb{R}^kf(a) \in \mathbb{R}^mgf(a) \in \mathbb{R}^kDfagf(a)gfa\textup{\textsf{cat}}*$ to the category of real vector spaces that sends a pointed manifold to its tangent space.}
\item Any commutative monoid $MM^{-} \colon \textup{\textsf{cat}}* \to \textup{\textsf{cat}}n_+ \in \textup{\textsf{cat}}*nM^{n+}M^nnMM^{0+}f \colon m+ \to n_+iM^f \colon M^m \to M^nM^mf^{-1}(i)M^fiM^nnM^{-}U \colon \textup{\textsf{cat}}_* \to \textup{\textsf{cat}}$.