In his Équations Différentielles à Points Singuliers Réguliers, Pierre Deligne affirms that
Let $X$ be a "nice" topological space. There exists a unique, up to isomorphism, locally constant sheaf of groups $\Pi_1(X)$ over $X$, endowed for all $x\in X$ with an isomorphism $$\Pi_1(X)_x\cong \pi_1(X,x),$$ and such that for every path $\gamma:[0,1]\to X$, the usual isomorphism between $\pi_1(X,\gamma(0))\cong \pi_1(X,\gamma(1))$ identifies via the isomorphism above to the natural isomorphism $\Pi_1(X)_{\gamma(0)}\cong \Pi_1(X)_{\gamma(1)}$.
(My translation of the original french.) How is this sheaf constructed? And what is its relation with the usual fundamental grupoid?
We want $\Pi_1(X)_x\simeq\pi_1(X,x)$, and to define the sheaf, the problem is how to glue. We can define $$\Pi_1(U):=\{(f_x)_{x\in U} \mid f_x\in\pi_1(X,x), \forall \gamma:[0,1]\rightarrow U, \iota_{\gamma}(f_{\gamma(0)}) = f_{\gamma(1)} \}$$ where $\iota_{\gamma}$ is the isomorphism induced by the path $\gamma$ as mentioned in the quote.
It's easy to verify $\Pi_1(U)$ is a sheaf. To show $\Pi_1(U)$ is locally constant (and has the desired germs), if $X$ is locally contractible (or at least semi-locally simply connected), then we can pick a neighborhood $U$ of any $x$, that is contractible, on which $\Pi_1(U)\simeq \pi_1(X,x)$ as for each choice $g\in \pi_1(X,x)$, we can define $f_y = \iota_{\gamma}(g)$ for any path $\gamma$ from $x$ to $y$, where $f_y$ doesn't depend on the choice of $\gamma$, as all $\gamma$'s are homotopic.