= Solution
At a point $x\in X$, all restrictions of the <constant presheaf of sets> are identities. Two representatives $(U,t)$ and $(V,t')$ give the same <germ> precisely when $t=t'$, so its <stalk> is canonically $T$. The disjoint union of the stalks is therefore the set $X\times T$.
The topology of the associated <étale space of a presheaf> has basic opens obtained from sections over open $U$. For the constant section $t$, that basic open is $U\times\{t\}$. These are exactly the basic opens of the product topology with $T$ discrete. Consequently the germ-space identification is a homeomorphism over $X$, and the projection
$$
\boxed{p:X\times T_{\mathrm{disc}}\longrightarrow X}
$$
is a local homeomorphism.
A continuous section over $V$ has the form $x\mapsto(x,f(x))$, where $f:V\to T_{\mathrm{disc}}$ is continuous, equivalently <locally constant>. Its sections form the <constant sheaf of sets>:
$$
\boxed{\Delta T(V)=\{f:V\to T:f\text{ is locally constant}\}.}
$$
The <sheaf gluing axiom> follows by gluing the functions; continuity is local. This includes $\Delta T(\varnothing)$ being a singleton, even when the constant presheaf's value on the empty open set was $T$. The construction is its <sheafification>.
A function $u:T\to T'$ acts by postcomposition on <locally constant> functions, defining a <functor> $\Delta:\mathbf{Set}\to\mathbf{Sh}(X)$. It preserves identities and composition.
To prove the <adjunction> with the <global sections functor>, let $F$ be a sheaf. A sheaf morphism $\lambda:\Delta T\to F$ gives a function
$$
T\longrightarrow F(X),\qquad
t\longmapsto\lambda_X(\text{constant function }t).
$$
Conversely, suppose global sections $s_t\in F(X)$ are prescribed for every $t\in T$. For a <locally constant> $f:V\to T$, its fibres $V_t=f^{-1}(t)$ are disjoint open sets covering $V$. Restrict $s_t$ to $V_t$ and glue these sections. They agree on intersections, which are empty, so unique gluing defines $\lambda_V(f)$. Restriction to smaller opens commutes with this construction, making $\lambda$ a sheaf morphism.
Applying the two constructions successively returns the original data: a <locally constant> function is locally one of the constant functions, and sheaf morphisms respect those restrictions and unique gluing. Naturality in $T$ and $F$ follows from postcomposition and restriction. Hence
$$
\boxed{\operatorname{Hom}_{\mathbf{Sh}(X)}(\Delta T,F)
\cong\operatorname{Set}(T,\Gamma F),\qquad \Delta\dashv\Gamma.}
$$
For an empty space the same proof works: all global sections sets are singletons, and the sheaf category is degenerate. Constant sheaves are generally <locally constant> rather than globally constant on disconnected opens.
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