Let $p:(E,e_0) \rightarrow(X,x_0)$ be a covering projection. Show that $p \sharp: \pi_{1}(E,e_0) \rightarrow \pi_{1}(X,x_0)$ is a monomorphism.
I was wondering here do I need to prove this is a homomorphism?
As I'm confused as I know you have to prove that it is an injection. Which is just by considering what is in the kernel of $\pi_{1}(E,e_0) \rightarrow \pi_{1}(X,x_0)$, something that is homotopic to the constant map, then you just lift the homotopy to the covering space to get something homotopic to the constant map in $(E,e_0)$.
However, is this enough or do you need to show homomorphism? How would you show a homomorphism? would you need to do concatenation of loops?