The fundamental groupoid of a topological space

Take the points of $X$ as objects and define $\operatorname{Hom}(x,y)$ to be the endpoint-preserving homotopy classes of paths from $x$ to $y$. For $\alpha:x\to y$ and $\beta:y\to z$, put $[\beta]\circ[\alpha]=[\alpha*\beta]$, and take the constant path at $x$ as $1_x$.

If $H$ deforms $\alpha$ to $\alpha'$ and $K$ deforms $\beta$ to $\beta'$ rel endpoints, paste $H(2s,t)$ for $s\le1/2$ to $K(2s-1,t)$ for $s\ge1/2$. The endpoint conditions agree along the seam, so the finite closed-pasting argument in [L2] gives an endpoint-preserving homotopy $\alpha*\beta\simeq\alpha'*\beta'$. Thus composition is well defined on classes.

Composing a path $\alpha$ with the straight-line homotopies from $s$ to $\max(0,2s-1)$ and to $\min(2s,1)$ proves the left and right identity laws. If $r(s)=2s$ for $s\le1/2$ and $r(s)=2-2s$ for $s\ge1/2$, then $\alpha((1-t)r(s))$ contracts $\alpha*\bar\alpha$ rel endpoints; the analogous formula $\alpha(t+(1-t)(1-r(s)))$ contracts $\bar\alpha*\alpha$. Hence reversal supplies a two-sided inverse class.

For three composable paths, let $\delta:[0,3]\to X$ traverse them successively. The two bracketings are $\delta\circ p$ and $\delta\circ q$, where $p(s)=4s$ for $s\le1/2$ and $p(s)=2s+1$ for $s\ge1/2$, while $q(s)=2s$ for $s\le1/2$ and $q(s)=4s-1$ for $s\ge1/2$. The formula $\delta((1-t)p(s)+tq(s))$ is an endpoint-preserving homotopy, so composition is associative on classes.

Steps 2.1, 3.1, and 2.2 make $\Pi_1(X)$ a groupoid. Its automorphisms at $x$ are the based-loop classes, but $[\beta]\circ[\alpha]=[\alpha*\beta]=[\alpha][\beta]$ in the traversal-order product of [L4]. Thus the identity on loop classes identifies $\operatorname{Aut}_{\Pi_1(X)}(x)$ with $\pi_1(X,x)^{\mathrm{op}}$. The inversion map $[\alpha]\mapsto[\bar\alpha]$ is therefore a canonical group isomorphism $\operatorname{Aut}_{\Pi_1(X)}(x)\cong\pi_1(X,x)$.