I am following Aluffi's Algebra: Chapter 0. In Chapter 1, Exercise 3.8, he gives the following definition of a subcategory.
A subcategory $\mathsf C'$ of a category $\mathsf C$ consists of a collection of objects of $\mathsf C$, with morphisms $\operatorname{Hom}_{\mathsf C'}(A, B)\subseteq\operatorname{Hom}_{\mathsf C}(A, B)$ for all objects $A$, $B$ in $\operatorname{Obj}(\mathsf C')$, such that identities and compositions in $\mathsf C$ make $\mathsf C'$ into a category.
Denoting the identities by $1_A\in\operatorname{Hom}_{\mathsf C}(A, A)$ and $1_A'\in\operatorname{Hom}_{\mathsf C'}(A, A)$, I thought at the first glance that the $1_A$ would naturally be in $\mathsf C'$ (for any object $A$ of $\mathsf C$). But I couldn't prove it from rest of the definition. And this led me to suspect that it is indeed a required condition.
Hence, there has to be an example where everything holds except that $1_A\notin \operatorname{Hom}_{\mathsf C'}(A, A)$. Can someone provide one?
Best Answer
A category with a single object is just a monoid $M$. Its subcategories, excluding the empty category, are just the submonoids of $M$. If one did not postulate that the identity maps must be the same, one would get the subsemigroups of $M$ which have an identity element.
In other words, you are essentially asking for an example of a monoid $M$ and a subsemigroup $S$ of $M$ which has a multiplicative unit but is not a submonoid of $M$. One such example is a linearly ordered set with a maximum element $1$ (such as the unit interval $[0, 1]$) where $x \cdot y = \min (x, y)$ and any subset with a maximum other than $1$ (such as the interval $[0, \frac{1}{2}]$).