I found this representation of a category.
I find the equation at the top quite redoundant.
My reasoning is that everything is depicted except for the identities. If I want to compose mother with firstChild I can only use what I find in the picture. That would leave me with the identity $id_{Mother}$.
Is the equation at the top necessary? Or is it just an aid?
The convention I've commonly seen is that that every path of morphisms composes to a distinct morphism unless otherwise implied by indicated compositions. So without the composition equation, the diagram would most commonly be read so that $\cdots\neq(\mathrm{mother}\circ\mathrm{firstChild})\circ(\mathrm{mother}\circ\mathrm{firstChild})\neq\mathrm{mother}\circ\mathrm{firstChild}\neq\mathrm{id}_{\mathrm{Mother}}$.
You can get around mentioning the equation if you say something like "category with two non-identity morphisms" which would force $\mathrm{mother}\circ\mathrm{firstChild}=\mathrm{id}_{\mathrm{Mother}}$ and $\mathrm{firstChild}\circ\mathrm{mother}=\mathrm{id}_{\mathrm{Child}}$. But either way, you have to say something to indicate how one is supposed to compose.