Let $G_1,G_2,H$ be groups such that there are injective homomorphisms $f_i\colon H\to G_i$ for $i=1,2$. We know that there are injective homomorphisms $\iota_i \colon G_i\to G_1*G_2$, where $G_1*G_2$ is the free product. We can then consider N as the smallest normal subgroup of $G_1*G_2$ that contains all the elements of the form $\iota_1(f_1(h))\iota_2(f_2(h^{-1}))$ with $h\in H$ and then the amalgamated product is defined as $G_1*_HG_2:=(G_1*G_2)/N$. With the definiton clear (please correct me in anything if I'm wrong), I'm trying to prove that there are injective homomorphisms $j_i:G_i\to G_1*_HG_2$, $i=1,2$.
My first attempt was to consider the natural projection $\pi:G_1*G_2\to (G_1*G_2)/N$ and then proposed $j_i:=\pi|_{\iota_i[G_i]}\circ\iota_i$, so basically, just restrict the natural projection to the copy of each $G_i$ in $G_1*G_2$ and compose with the already existent inclusions.
Now, for the sake of having a simpler notation, let's call $G_i^{*}=\iota_i[G_i]$. Since the inclusions to the free product are already injective, it suffices to prove that the restricted natural projection behaves injectively. We have that $ker(\pi|_{G_i^*})=G_i^*\cap N$, so proving that it is injective is equivalent to each $G_i^*$ having trivial intersection with $N$, so let $x\in G_i^*\cap N$. The struggle comes with how general the definitons are, because for example, if $N$ was the subgroup generated by the same elements as before, then one would at least have an idea of how its elements would look like. However, we are taking the intersection of all normal subgroups that contain those elements, and so the only thing I know is that $x$ belongs to each of these subgroups. Also, given an arbitrary $g$, I would want to prove that $gxg^{-1}=e$, but the only thing I know is that it is again an element of $N$.
I know maybe I haven't achieved much progress, but I really don't know how to proceed, as I'm very new to this kind of topics. Any help is greatly appreciated and thanked for.
P.S. Is this topic not a little too advanced for an introductory group theory course? I feel as though my professor was teaching us topics that way are beyond the ones that would usually be teached in a first class of group theory.