= Solution
For the covariant category in this part, put $\mathcal E=[\mathcal C,\mathbf{Set}]$ and $r_c=\mathcal C(c,-)$. Its global-sections <functor> is $\Gamma=\operatorname{Hom}(1,-)$, the limit of the diagram. If $\Gamma$ is also an inverse image, it is a <left adjoint> and preserves all small <colimits>.
Consider the canonical pointwise-surjective map
$$
q:\coprod_{c\in\mathcal C}r_c\longrightarrow1.
$$
At $d$, the summand $r_d(d)$ contains $1_d$, so $q$ is an <epimorphism>. In a <presheaf topos> <epimorphisms> are regular, and a colimit-preserving <functor> preserves their coequalizer presentations. Thus $\Gamma q$ is surjective. Since $\Gamma$ preserves <coproducts>, some summand has a global element, yielding a section $s:1\to r_c$ of its unique map $r_c\to1$. \b[The constant singleton <functor> is a retract of a covariant representable.]
Conversely, if $1$ is such a <retract in a category>, $\operatorname{Hom}(1,-)$ is a retract of $\operatorname{Hom}(r_c,-)$, which is evaluation at $c$ by the <Yoneda lemma>. Evaluation preserves <colimits> pointwise. The <colimit> comparison for a retract <functor> is a retract of the evaluation comparison; a retract of an isomorphism is an isomorphism, so $\Gamma$ preserves all small <colimits>. It also preserves <finite limits> because it is representable. A <right adjoint> can be constructed by
$$
R(S)(d)=\operatorname{Set}(\Gamma r_d,S).
$$
For $h:d\to d'$, precomposition gives $r_{d'}\to r_d$, and applying $\Gamma$ and then $\operatorname{Set}(-,S)$ gives $R(S)(d)\to R(S)(d')$. The presentation of every <functor> as a <colimit> of representables gives $\operatorname{Hom}(F,R(S))\cong\operatorname{Set}(\Gamma F,S)$. Hence $\Gamma\dashv R$, so $\Gamma$ is an inverse image and the topos is local.
The retract condition has a precise interpretation as the <initial object criterion for a covariant local topos>. Write the section as a natural family $u_d:c\to d$. Naturality says $h u_d=u_{d'}$ for every $h:d\to d'$. Put $e=u_c$. Then $e^2=e$ and every $f:c\to d$ satisfies $fe=u_d$. In the <idempotent completion> of $\mathcal C$, the object $(c,e)$ is initial: an arrow from it to $(d,k)$ must satisfy $v=kv e$, and the identities force $v=u_d$; this arrow exists because $k u_d=u_d$ and $u_d e=u_d$.
Conversely an <initial object> $(c,e)$ in the <idempotent completion> gives that natural family and the retract. Therefore
$$
\boxed{[\mathcal C,\mathbf{Set}]\text{ is local}
\ \Longleftrightarrow\
\operatorname{Kar}(\mathcal C)\text{ has an initial object}.}
$$
If $\mathcal C$ is already <idempotent-complete>, this is equivalent to an <initial object> in $\mathcal C$ itself, and $\Gamma$ is evaluation there. The word is initial, not terminal, because this question uses covariant <functors>.
One cannot omit <idempotent completion> for a general small category. Take the one-object category of the <monoid> $\{1,0\}$ with absorbing zero. It has no <initial object> because its endomorphism set has two elements, but the natural family $u=0$ satisfies $m0=0$ for every $m$. Its covariant <functor> category is local. Concretely, a <functor> is a set with an idempotent operator, and its global sections are the fixed points of that operator.
For a <topological space> $X$, there is an equally explicit criterion. If $\mathbf{Sh}(X)$ is local, apply $\Gamma$ to the <epimorphism> $\coprod_iU_i\to1$ associated with any open cover $X=\bigcup_iU_i$, regarding the opens as <subterminal sheaves>. Since $\Gamma U_i$ is a singleton exactly when $U_i=X$, and otherwise empty, <coproduct> and <epimorphism> preservation force some $U_i=X$.
Thus every open cover of $X$ contains $X$ itself. Conversely this <open cover criterion for a local sheaf topos> condition is equivalent to existence of a point $x$ whose only open neighborhood is $X$: the union of all proper open sets cannot be all of $X$, so choose $x$ outside it. Such a point plainly forces any cover to contain $X$. Empty spaces fail the condition.
At this point, the <stalk functor> $F\mapsto F_x$ is simply $F(X)=\Gamma F$, since there is only one open neighborhood. Its <right adjoint> is the <skyscraper sheaf of sets>
$$
R(S)(V)=
\begin{cases}S,&x\in V,\\\{*\},&x\notin V.\end{cases}
$$
A morphism $F\to R(S)$ is determined exactly by its map $F(X)\to S$, because all proper opens omit $x$. Hence $\Gamma\dashv R$, and $\Gamma$ preserves <finite limits>, proving locality. Consequently
$$
\boxed{\mathbf{Sh}(X)\text{ is local}
\ \Longleftrightarrow\
\exists x\in X\text{ with no proper open neighborhood}.}
$$
In a $T_0$ space this distinguished point is unique and closed. In a $T_1$ space the condition forces $X$ to be a singleton. Indistinguishable points in a non-$T_0$ space can all have this property; uniqueness of an actual point is not part of the general criterion.
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