Dislaimer: I know very little about this, so if parts of my question don't make sense, please feel free to edit in a way that does, or ask me to clarify.
Let $\mathcal{M}_{1,2}$ be the moduli space of genus 1 curves (plane curves with base field $\mathbb{C}$) with 2 marked points. I think there is a map (thought of as addition in a sense like addition on an elliptic curve) that is \begin{align*} \mathcal{M}_{1,2} \times \mathcal{M}_{1,2} &\to \mathcal{M}_{1,1} \\ (C,p_1,p_2), (C,p_1,p_3) &\mapsto (C,p_2+p_3 - p_1) \end{align*} where $p_1$ is sort of an identity element. My question is: is this correct? If so, how should I think about it, and where can I read more about it?