The constant sheaf is the sheaf of locally constant functions

Define a presheaf map $\phi :A_{\mathrm{p}\mathrm{t}}\to {\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ by ${\phi }_U(a):=(x\mapsto a)$, the constant function with value $a$ on $U$. This is a morphism of presheaves in the sense of [F1]: for $V\subseteq U$ and $a\in A$ the restriction of the constant function with value $a$ on $U$ to $V$ is again the constant function with value $a$, and ${\phi }_V(a)$ is that same function, so ${\phi }_V(a{\mathrm{\mid }}_V)={\phi }_U(a){\mathrm{\mid }}_V$, both sides being constant with value $a$; here $a{\mathrm{\mid }}_V=a$ because all restriction maps of $A_{\mathrm{p}\mathrm{t}}$ are the identity.

The stalk of the constant presheaf at $x\in X$ is $A$: by [F5] it is the colimit of the diagram which is constant with value $A$ on the filtered category of open neighbourhoods of $x$, and a cocone from that diagram to a set $S$ is the same thing as a single map $A\to S$ (all the structure maps of $A_{\mathrm{p}\mathrm{t}}$ are identities, so the compatibility conditions are automatic); by the explicit description of a colimit [F6] the identity of $A$ exhibits $A$ as a colimit, so the canonical map $A\to (A_{\mathrm{p}\mathrm{t}}{)}_x$, $a\mapsto$ the class of the constant section $a$ over $X$, is a bijection.

By [F2] the presheaf ${\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ is a sheaf of sets, so by the universal property [F3] the morphism $\phi$ factors uniquely through the sheafification map $\eta :A_{\mathrm{p}\mathrm{t}}\to A_X$: there is exactly one morphism of sheaves of sets $\theta :A_X\to {\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ with $\theta \circ \eta =\phi$. In particular ${\theta }_U({\eta }_U(a))={\phi }_U(a)$ is the constant function with value $a$, which is the compatibility asserted in clause 2.

Under the identifications of [step 1.2] and of the evaluation bijection of [F2], the induced map ${\phi }_x:(A_{\mathrm{p}\mathrm{t}}{)}_x\to {\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c},x}$ is the identity of $A$: the element $a$ corresponds to the class of the constant section with value $a$ over $X$, its image under $\phi$ is the germ of the constant function with value $a$, and evaluation at $x$ returns $a$. Hence ${\phi }_x$ is a bijection for every $x\in X$.

For every $x\in X$ the map ${\eta }_x:(A_{\mathrm{p}\mathrm{t}}{)}_x\to (A_X{)}_x$ is a bijection by [F4], and ${\phi }_x={\theta }_x\circ {\eta }_x$ by [step 2.1]; since ${\phi }_x$ is a bijection by [step 2.2], the map ${\theta }_x$ is a bijection as well, being the composite of the inverse of ${\eta }_x$ with ${\phi }_x$.

By [step 3.1] every induced map of $\theta$ on stalks is a bijection, so $\theta :A_X\to {\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ is an isomorphism of sheaves of sets by [F7]. Together with [step 2.1] this is clauses 1 and 2 of the statement.

Suppose now that $A$ is an abelian group. Pointwise addition makes $A_{\mathrm{p}\mathrm{t}}$ a presheaf of abelian groups, all of whose restriction maps are the identity, and makes ${\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ a sheaf of abelian groups: the sum and the negative of locally constant functions are locally constant, since on a neighbourhood where each summand is constant the sum is constant, and restrictions are the corresponding group homomorphisms [F8]. Each ${\phi }_U$ is a group homomorphism because constant functions add pointwise. Transport the group operation of ${\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}(U)$ to $A_X(U)$ along the bijection ${\theta }_U$ of [step 4.1], declaring $s+t:={\theta }_U^{-1}({\theta }_U(s)+{\theta }_U(t))$ for $s,t\in A_X(U)$: this makes every ${\theta }_U$ a group isomorphism, and the restriction maps of $A_X$ are group homomorphisms because they are conjugate through $\theta$ to the restriction maps of ${\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$, which are homomorphisms, so $A_X$ is a sheaf of abelian groups; moreover ${\eta }_U={\theta }_U^{-1}\circ {\phi }_U$ is then a group homomorphism, since it is a composite of group homomorphisms and the inverse of one. This is clause 3, and the proof is complete. ∎