I am currently studying categories without products. I have found that the category of fields has no products Examples of a categories without products, in the proof an endomorphism appears and while I was trying to understand it, I came up with the question Are all endomorphisms identities in a category?
I have no idea how to answer that question. Can someone help me?
Of course not. Even automorphisms need not be identities (an endomorphism that is also an isomorphism is an automorphism). As we all know, a group is the set of isomorphisms in a groupoid $\mathsf G$ with a single object, endowed with the operation of composition of morphisms: $\text{Aut}_{\mathsf G}(∗)$. Only in a trivial group are all the automorphisms identities.