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Every restriction is the identity. Its stalk at each point is $T$, and its étale space of a presheaf is $X\times T$ with $T$ discrete. It is generally not a sheaf; in particular its value on the empty open set is $T$. Its sheafification is the constant sheaf of sets.

At a point $x\in X$, all restrictions of the constant presheaf of sets are identities. Two representatives $(U,t)$ and $(V,t^{\prime })$ give the same germ precisely when $t=t^{\prime }$, 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 projectionis 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: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 $u:T\to T^{\prime }$ 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 functionConversely, 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. HenceFor 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.