Let $\varphi:G\to H$ and $\psi: H\to I$ and $\chi:I\to J$ be homomorphisms of the groups $G,H,I,J$. I want to prove that homomorphisms are associative, that is to say $\varphi\circ (\psi\circ \chi) = (\varphi\circ\psi)\circ \chi$. My proof is as follows:
All homomorphisms are functions. All functions are associative. Therefore all homomorphisms are associative.
However, my tutor claimed that this was a weird and over-complicated proof, and that homomorphisms are just associative by definition (and that most mathematicians would be "boggled at the weird gyrations" that I made in my proof). I wanted to see if I have some kind of profoundly weird or mistaken understanding here. I claimed that homomorphisms are not defined to be associative, as I don't see that in the Wikipedia article or any textbook. (https://en.wikipedia.org/wiki/Homomorphism)
(Of course my proof relies on the principle that all functions are associative, which I believe for the purpose of this proof I can assume.)