The following question is related to this other question. In the question I have to show the exactness of $$\pi_1(A,x_0) \overset{i_*}{\longmapsto} \pi_1(X,x_0)\overset{j_*}{\longmapsto} \pi_1(X,A,x_0)$$
In particular I'd like to prove that $j_*^{-1}(c_{x_0}) \subseteq i_*(\pi_1(A,x_0))$ where $c_{x_0}$ is the costant path in $x_0$.
So taken an element $[\gamma] \in j_*^{-1}(c_{x_0})$ this is represented by a path $\gamma : I \longmapsto X$ such that $j \circ \gamma$ is homotopic to $c_{x_0}$, i.e exists $H : (I,\partial I, 0) \longmapsto (X,A,x_0)$ such that $H_0 = j \circ \gamma, H_1 = c_{x_0}$.
Now take $\sigma := H(1,t)$ is a loop $A$ and the book says that as the question linked that the image under $i_*$ of this loop goes to the class of $[\gamma]$.
What I don't understand is: Is it true that $[j \circ \gamma] = [\gamma]$ ? Is this true only in $\pi_1(X,A,x_0)?$ it seems always an abuse of notation saying $[j \circ \gamma] = [\gamma]$ and I don't understand the fact that we found an homotopy given by definition between $j \circ \gamma$ and the costant loop in $x_0$, so here $\gamma$ doesn't seem to play a role so why do we know that $[\gamma]$ is in the image of $i_*$ if the homotopy is between $j \circ \gamma$ and $\gamma?$
Any help to clarify this detail would be appreciated, references or solutions included.