= Solution
Work in the based CW <homotopy category>. A homotopy-invariant <functor> depends on a map only through its based <homotopy class>. In a sequence of <pointed sets>, exactness at the middle means that the preceding image equals the inverse image of the distinguished point; it does not imply injectivity or surjectivity at the ends. For <groups> this is the familiar image-equals-kernel condition. A <half-exact functor> preserves such middle exactness on the appropriate elementary sequence.
For a contravariant <functor> $F$, the relevant elementary sequence is a <cofibration> $A\hookrightarrow X\to X/A$. Half-exactness is
$$
F(X/A)\longrightarrow F(X)\longrightarrow F(A)
$$
exact at $F(X)$. One should not silently turn this into a short exact sequence. In particular, arbitrary <group> actions and <pointed sets> at low degrees are not interchangeable with <abelian groups>.
The basic contravariant example is $F_Y(X)=[X,Y]_*$, often called a cohomotopy-type <functor>. Suppose a map $f:X\to Y$ becomes <null-homotopic> on $A$. A chosen null-homotopy extends it over the cone attached to $A$, giving a map from the <topological mapping cone> to $Y$. Conversely every such extension restricts to a <null-homotopic> map on $A$. Since the <topological mapping cone> of a <subcomplex> inclusion models $X/A$, this proves half-exactness, rather than merely asserting it.
Repeating the <topological mapping cone> construction gives the <Puppe sequence>
$$
A\to X\to C_i\to\Sigma A\to\Sigma X\to\cdots.
$$
Contravariant mapping into $Y$ gives
$$
\cdots\to[\Sigma X,Y]_*\to[\Sigma A,Y]_*\to[C_i,Y]_*\to[X,Y]_*\to[A,Y]_*.
$$
This is exact as <pointed sets>. An <H-group> structure on $Y$, for example a <loop space> with path concatenation, provides <group> structures; repeated looping gives <abelian group> structures in the usual ranges. The <suspension-loop adjunction> $[\Sigma X,Y]_*\cong[X,\Omega Y]_*$ explains the shifted terms and their compatibility.
The covariant counterpart is $H_T(Y)=[T,Y]_*$, with ordinary <homotopy groups> obtained from $T=S^n$. Its natural elementary sequence is a <homotopy fiber sequence> $F\to E\to B$, not an arbitrary <cofibration>. A map $a:T\to E$ whose composite to $B$ is <null-homotopic>, together with that null-homotopy, determines a map into the <homotopy fibre>. Conversely a fibre map projects to such a null composite. This proves exactness of $[T,F]_*\to[T,E]_*\to[T,B]_*$ at the middle. Adding the loop terms produces the <homotopy> long exact sequence. For a <Serre fibration>, its ordinary fibre supplies the same <homotopy> information.
This variance distinction matters. Include one <circle>, generated by $a$, in $S^1\vee S^1$, with other generator $b$. The quotient is the second <circle>. The kernel of $\pi_1(S^1\vee S^1)\to\pi_1(S^1)$ is the <normal closure> of $a$, which contains $bab^{-1}$. The image of the included <circle> is only the <cyclic subgroup> generated by $a$, and does not contain that reduced word. Thus covariant $\pi_1$ is not half-exact on this <cofibration>. Fibre and cofibre statements cannot be swapped just by reversing notation.
Reduced <cohomology> gives group-valued contravariant examples. For an abelian coefficient <group> $G$, the degree-$q$ <functor> is represented by an <Eilenberg–MacLane space>:
$$
\widetilde H^q(X;G)\cong[X,K(G,q)]_*.
$$
Its <cofibration> exact sequence continues as $\cdots\to\widetilde H^{q-1}(A)\to\widetilde H^q(X/A)\to\widetilde H^q(X)\to\widetilde H^q(A)\to\cdots$. Ordinary <homotopy groups> are instead covariant, with decreasing <group> degree across a fibre connecting map. These are the two complementary representable patterns, maps out of a test object and maps into a representing object.
<Brown's representability theorem> makes the contravariant pattern structural. A reduced contravariant <homotopy> <functor> on based <CW complexes> is represented by a <CW complex> if it converts arbitrary wedges to products and satisfies the Mayer-Vietoris weak-pullback gluing axiom. The latter says that compatible classes on two subcomplexes extend to their union. It contains more information than the single kernel statement of half-exactness. Consequently half-exactness alone is not a representability criterion. In a representing <functor> these axioms are natural: a map on a wedge is a family of based maps, while compatible maps on an intersection can be adjusted by a <homotopy> and glued across a <homotopy pushout>.
A reduced <generalized cohomology theory> with the usual wedge axiom supplies these axioms degree by degree. Its representing spaces $E_q$ have $E_q\simeq\Omega E_{q+1}$ by suspension compatibility, assembling an <Omega-spectrum>. Thus generalized <cohomology> can be expressed as $E^q(X)=[\Sigma^\infty X,\Sigma^qE]_{\mathrm{st}}$, while represented <homology> is $E_q(X)=\pi_q(E\wedge\Sigma^\infty X)$. The appearance of mapping into a spectrum in one case and smashing with it in the other is another form of the same variance distinction.
There is a literal duality after stabilization and a finiteness restriction. Here a finite based <CW complex> is identified with its <suspension spectrum>. For a <finite spectrum> $X$, let $DX=F(X,\mathbb S)$ be its <Spanier-Whitehead dual>. Then $D^2X\simeq X$ and $[X,Y]_{\mathrm{st}}\cong[DY,DX]_{\mathrm{st}}$ for <finite spectra>. Duality reverses a cofibre triangle into a fibre triangle. In particular,
$$
\widetilde E^q(X)\cong\pi_{-q}(E\wedge DX),\qquad \widetilde E_q(X)\cong[\Sigma^qDX,E]_{\mathrm{st}}.
$$
This turns a <cohomology> calculation on a finite object into a <homology> calculation on its dual, and vice versa. It is <Spanier-Whitehead duality>, not a claim that the full category of ordinary spaces is self-dual.
\b[Half-exactness is the local image–kernel property; fibres and cofibres, representability, and finite stable duality explain how that property organizes <homotopy> and <cohomology>.] The distinctions between variance, <pointed sets> and <groups>, and finite stable objects and arbitrary spaces are essential to using the duality correctly.
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