Any isomorphism $f: x \rightarrow x$ in a category has to be the identity morphism?

Question

Let $C$ be a category and $f: x \rightarrow x$ be an isomorphism. Is it true that $f$ must be $id_x$? If no, what would be a good counterexample?

Answer 1

If $C=\mathcal{Set}$, the category of sets, and $x=\{a,b\}$ then we have two isomorphisms $\{a,b\}\to\{a,b\},$ one the identity and one which swaps $a,b.$

In general, in $\mathcal{Set},$ the isomorphisms $x\to x$ are the permutations of the set $x.$

Answer 2

No. See, for instance, the category of vector spaces over some field $k$. And almost any category of common usage in algebra/geometry.

Answer 3

No. A single group is a category with one object and right multiplication by an element is a morphism $x \to x$.

You can visualize a group as a daisy with all the elements as petals looping back to the floral disc.