The sheaf of all functions to an abelian group is flasque

For open $U\subseteq X$ let $\mathcal{F}(U)=\mathrm{Map}(U,A)$ be the set of all functions $U\to A$, and for open $U\subseteq V$ let ${\rho }_U^V(f):=f{\mathrm{\mid }}_U$; both $U$ and $V$ are open sets of the topology of [F1]. Restriction of functions satisfies ${\rho }_U^U(f)=f$ and ${\rho }_W^U(f)={\rho }_W^V({\rho }_V^U(f))$ for $W\subseteq V\subseteq U$, since both sides send $x\in W$ to $f(x)$; by [F2] this makes $\mathcal{F}$ a presheaf of sets on $X$. It is a presheaf of abelian groups under pointwise addition $(f+g)(x):=f(x)+g(x)$: the pointwise sum of two functions $U\to A$ is a function $U\to A$ [F5], addition is associative and commutative and has the constant zero function as identity because $A$ is an abelian group, and each ${\rho }_U^V$ is a group homomorphism since restrictions are computed valuewise.

$\mathcal{F}$ satisfies the two sheaf conditions of [F3]. Locality: if $f,g\in \mathcal{F}(U)$ and $f{\mathrm{\mid }}_{U_i}=g{\mathrm{\mid }}_{U_i}$ for all $i$ in an open cover $U={\bigcup }_{i\in I}{U}_i$, then for every $x\in U$ there is an $i$ with $x\in U_i$, and $f(x)=f{\mathrm{\mid }}_{U_i}(x)=g{\mathrm{\mid }}_{U_i}(x)=g(x)$, so $f=g$. Gluing: let $f_i\in \mathcal{F}(U_i)$ satisfy $f_i{\mathrm{\mid }}_{U_i\cap U_j}=f_j{\mathrm{\mid }}_{U_i\cap U_j}$ for all $i,j$, and form the relation $f:=\{(x,v):x\in U,v\in A,\text{ there is }i\in I\text{ with }x\in U_i\text{ and }{f}_i(x)=v\}\mathrm{.}$ If $(x,v)$ and $(x,v^{\mathrm{\prime }})$ belong to $f$, witnessed by indices $i,j$ with $x\in U_i\cap U_j$, then $v=f_i(x)=f_i{\mathrm{\mid }}_{U_i\cap U_j}(x)=f_j{\mathrm{\mid }}_{U_i\cap U_j}(x)=f_j(x)=v^{\mathrm{\prime }}$, so the value is unique and $f$ is a function [F5] with domain $U$: every $x\in U$ lies in some $U_i$, giving $(x,f_i(x))\in f$. By construction $f{\mathrm{\mid }}_{U_i}=f_i$ for every $i$, so compatible families glue; by [F3] the presheaf $\mathcal{F}$ is a sheaf, and with the pointwise group structure of [step 1.1] it is a sheaf of abelian groups, the group operations being computed valuewise and the glued section unique.

Let $U\subseteq V$ be open and let $f\in \mathcal{F}(U)$. Since $V$ is the disjoint union of $U$ and $V\setminus U$, the rule $g(x):=\{\begin{array}{ll}f(x),& x\in U,\\ 0_A,& x\in V\setminus U,\end{array}$ defines a function $g:V\to A$ [F5], because the two cases are exhaustive and mutually exclusive and the values are prescribed by the given data; here $0_A$ is the zero element of the abelian group $A$. Its restriction to $U$ is ${\rho }_U^V(g)=f$. Hence every element of $\mathcal{F}(U)$ has a preimage under ${\rho }_U^V$, that is, ${\rho }_U^V$ is surjective. As $U\subseteq V$ were arbitrary open subsets, all restriction maps of $\mathcal{F}$ are surjective, and by [F4] the sheaf $\mathcal{F}$ is flasque.

Collecting the two assertions: $\mathcal{F}$ is a sheaf of abelian groups on $X$ by [step 1.1] and [step 2.1], and it is flasque by [step 3.1], because every restriction of a function to a smaller open set has the canonical extension by the zero element of $A$ described there. In particular the statement holds for every abelian group $A$ and every topological space $X$, with $\mathcal{F}(\mathrm{\varnothing })=\mathrm{Map}(\mathrm{\varnothing },A)$ the one-element group. No choice principle is used anywhere: the extension of [step 3.1] is given by an explicit two-case formula and the glued function of [step 2.1] is defined by a relation whose values are unique, so that no index or point is selected and the item declares no choice principle. ∎