Group isomorphism between identity morphisms of fundamental groupoid and fundamental group of topological space

Question

For any topological space $X$, I have defined $C=\Pi_1(X)$ by objects $\text{Ob}(C)=X$ and morphisms $\text{Hom}_C(x,y)=\text{HPath}(x,y)$, where $\text{HPath}(x,y)$ denotes the set of homotopy classes $[\gamma]$ of continuous maps $\gamma:[0,1]\longrightarrow X$ between $x$ and $y$ and where composition is given by concatenation and reparametrisation of representative maps. I have shown in detail that this is a category where all morphims are invertible, ie. a groupoid. Now I need to verify that there exists a group isomorphism between $\text{Hom}_C(x,x)$ and $\pi_1(X,x)$ for all $x$ in $X$, where $\pi_1(X,x)$ denotes the fundamental group of $X$ at point $x$. Since I am not used with working with the fundamental group, I am struggling here. Can I explicitly construct a map between both and then show that it is a group homomorphism that is bijective? Or what is another basic way to prove it? Thanks for your help!