I am currently trying to learn about the Eichler-Shimura isomorphism, and all the definitions seems to be somewhat cumbersome and hard to commit to memory as intuitive. On the other hand, the following map feels very intuitive/simple:
$$f(z)\to j^{k-2}(M,z)f^{(k-1)}(Mz)-f^{(k-1)}(z)$$
Here $f\in\mathcal{M}_k(\Gamma)$, $j(M,z)=cz+d$, and $f^{(k-1)}(Mz)$ denotes $k-1$ fold integration of $f$.
Since the choice of constants is arbitrary the term "$j^{k-2}(M,z)f^{(k-1)}(Mz)-f^{(k-1)}(z)$" is only defined up to a term $p^M-p$ where $p$ is a polynomial of degree $\leq k-2$ and $M$ is the action $f\to j^{k-2}(M,z)f(Mz)$. These terms are exactly the coboundaries in $H^1(\Gamma,V(k))$, so assuming that $j^{k-2}(M,z)f^{(k-1)}(Mz)-f^{(k-1)}(z)$ has the cocycle property then we have a natural map $\mathcal{M}_k(\Gamma)\to H^1(\Gamma,V(k))$.
To prove that $j^{k-2}(M,z)f^{(k-1)}(Mz)-f^{(k-1)}(z)$ has the cocyle property we simply note that the $(k-1)$-fold differentiation of $j^{k-2}(M,z)f^{(k-1)}(Mz)$ is exactly equal to $j^{-k}(M,z)f(Mz)$ (this can be seen through a simple induction) and thus the modularity property (up to polynomials) transfers.
I'm not quite sure how this extends to a map $\mathcal{M}_k(\Gamma)\oplus \overline{S_k(\Gamma)} \to H^1(\Gamma,V(k))$; one could $(f,g)$ to the sum of the natural embeddings of $f$ and the natural embedding of $g$ but this feels arbitrary. Moreover, I am not sure if I can prove Eichler-Shimura using this morphism. Is it equivalent to the already known ones? Is there some reference where people define this morphism? I find it hard to believe that I am the first one to come up with this interpretation of Eichler-Shimura.