Question on functors please i need help,how to prove that 
"the functor (covariant) "fundamental group", of the category of pointed topological spaces in the category of groups" is really a functor 
What i must do to prove it ?
please, thank you.
 A: I'll put the definitions here so that you can hopefully work through the problem yourself.
Let $\mathcal{C}$ and $\mathcal{D}$ be categories. A functor $F\colon\mathcal{C}\rightarrow\mathcal{D}$ is an assignment, to every object $A\in \mbox{Obj}(\mathcal{C})$ an object $F(A)\in \mbox{Obj}(\mathcal{D})$ and to every morphism $f\colon A\rightarrow A'\in\mbox{Mor}(\mathcal{C})$, a morphism $F(f)\colon F(A)\rightarrow F(A')\in\mbox{Mor}(\mathcal{D})$ such that for composable morphisms $f,g$, we have $$F(f\circ g)=F(f)\circ F(g)$$ and for the identities $\mbox{Id}_A$, we have $F(\mbox{Id}_A)=\mbox{Id}_{F(A)}$.
Let $X$ be a topological space with base point $x_0$ and let $P=\{\gamma\colon[0,1]\rightarrow X\mid \gamma(0)=\gamma(1)=x_0\}$ be the set of all loops in $X$ based at $x_0$. We define the equivalence relation $\sim$ on the set $P$ by $\gamma\sim\gamma'$ is $\gamma$ and $\gamma'$ are homotopic functions relative to their end points. We then define a new a new set $$\pi_1(X,x_0)=P/{\sim}$$ which is the set of classes of homotopic relative to end points loops in $X$ based at $x_0$. The set $\pi_1(X,x_0)$ has a natural group structure $\#$ given by $[\gamma]\#[\gamma']=[\gamma\wedge\gamma']$ where here $\gamma\wedge\gamma'\colon[0,1]\rightarrow X$ is the loop defined by $$(\gamma\wedge\gamma')(t)=\begin{cases}\gamma(2t)&\mbox{ if }t\in\left[0,\frac{1}{2}\right]\\ \gamma'(2t-1)&\mbox{ if } t\in\left[\frac{1}{2},1\right]\end{cases}$$ and we call the group $(\pi_1(X,x_0),\#)$ the fundamental group of $X$, based at $x_0$.
To every basepoint preserving map $f\colon (X,x_0)\rightarrow (X',x_0')$ (so $f(x_0)=x_0'$), there is a natural induced homomorphism, which we denote by $\pi_1(f)$, on the level of fundamental groups defined by $$\pi_1(f)([\gamma])=[f\circ\gamma].$$
The question you are being asked is to show that $\pi_1\colon\mathbf{Top}_*\rightarrow\mathbf{Grp}$ which assigns to based spaces their fundamental group, and to the basepoint preserving maps between pointed spaces their induced homomorphisms on the level of groups, is a functor. To do this, you need to verify that the conditions mentioned at the start of this answer for the definition of a functor are all satisfied.
