Locally constant functions form a sheaf and have constant stalks

Restriction preserves local constancy. If two locally constant functions on $U$ agree on an open cover, then they agree pointwise on $U$. If locally constant functions $f_i:U_i\to A$ are compatible on an open cover of $U$, the pointwise glued function $f$ is well defined and locally constant because near any point it agrees with one of the local functions $f_i$. Hence [L1] holds and ${\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ is a sheaf.

Define ${ϵ}_x:{\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c},x}\to A$ by ${ϵ}_x([U,f])=f(x)$. This is well defined because equal germs agree on some neighbourhood of $x$, hence have the same value at $x$. Every $a\in A$ is the value at $x$ of the constant function $a$ on any neighbourhood of $x$, so ${ϵ}_x$ is surjective. If ${ϵ}_x([U,f])={ϵ}_x([V,g])$, then $f(x)=g(x)$. Since both functions are locally constant, there is a neighbourhood $W$ of $x$ on which $f$ and $g$ are both constantly this common value, so $[U,f]=[V,g]$. Therefore ${ϵ}_x$ is bijective.