fundamental groupoid
Definition 1.
Given a topological space^{} $X$ the fundamental groupoid^{} ${\mathrm{\Pi}}_{1}(X)$ of $X$ is defined as follows:

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The objects of ${\mathrm{\Pi}}_{1}(X)$ are the points of $X$
$$\mathrm{Obj}({\mathrm{\Pi}}_{1}(X))=X,$$ 
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morphisms^{} are homotopy classes of paths “rel endpoints” that is
$${\mathrm{Hom}}_{{\mathrm{\Pi}}_{1}(X)}(x,y)=\mathrm{Paths}(x,y)/\sim ,$$ where, $\sim $ denotes homotopy^{} rel endpoints, and,

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composition^{} of morphisms is defined via concatenation of paths.
It is easily checked that the above defined category is indeed a groupoid^{} with the inverse^{} of (a morphism represented by) a path being (the homotopy class of) the “reverse” path. Notice that for $x\in X$, the group of automorphisms^{} of $x$ is the fundamental group^{} of $X$ with basepoint $x$,
$${\mathrm{Hom}}_{{\mathrm{\Pi}}_{1}(X)}(x,x)={\pi}_{1}(X,x).$$ 
Definition 2.
Let $f:X\to Y$ be a continuous function between two topological spaces. Then there is an induced functor^{}
$${\mathrm{\Pi}}_{1}(f):{\mathrm{\Pi}}_{1}(X)\to {\mathrm{\Pi}}_{1}(Y)$$ 
defined as follows

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on objects ${\mathrm{\Pi}}_{1}(f)$ is just $f$,

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on morphisms ${\mathrm{\Pi}}_{1}(f)$ is given by “composing with $f$”, that is if $\alpha :I\to $ $X$ is a path representing the morphism $[\alpha ]:x\to y$ then a representative of ${\mathrm{\Pi}}_{1}(f)([\alpha ]):f(x)\to f(y)$ is determined by the following commutative diagram^{}