$\newcommand{\cat}[1]{\mathbf{#1}}$ $\newcommand{\Ob}{\text{Ob}}$ $\newcommand{\Mor}{\text{Mor}}$ I'm following Riehl's category theory in context, one of her lemmas in section 1.1 is the following.
Lemma 1.1.13: Any category $\cat{C}$ contains a maximal groupoid, the subcategory containing all of the objects, and only those morphisms that are isomorphisms.
Proof Attempt: Let $\cat{D}$ denote the maximal groupoid. It is obvious that $\Mor \cat{D}$ is a subcollection of $\Mor \cat{C}$ and $\Ob \cat{C} = \Ob \cat{D}$, i.e it's objects and morphisms are subcollections of $\cat{C}$. Next for any morphism $f \in \Mor \cat{D}$ it's associated domain and codomain objects are obviously in $\Ob \cat{D}$. Then if $x \in \Ob \cat{D}$ we know that $1_x \in \cat{D}$ since identity arrows are obviously isomorphisms. Lastly if $a \xrightarrow{f} b \xrightarrow{g} c$ are arrows in $\cat{D}$ then $gf \in \Mor \cat{D}$ since it has inverse $f^{-1}g^{-1}:c \rightarrow a$.
Question 1: Is this proof correct?
If so how can I deal with the following issue. Riehl leaves the proof of this lemma to the exercises section, but in that section she writes the following.
Exercise 1.1.ii: Let $\cat{C}$ be a category. Show that the collection of isomorphisms in $\cat{C}$ defines a subcategory, the maximal groupoid inside $\cat{C}$.
Question 2: I'm having a conceptual issue because in the exercise she no longer has anything about the objects of the maximal groupoid. From a novice point of view, I feel as though it's presenting two different definitions for the maximal groupoid. First, the subcategory consisting of all objects and only isomorphisms, and then simply the subcollection of all isomorphisms in $\cat{C}$.
I suppose I understand that if you have the collection of all isomorphisms, then if it is going to be a subcategory; meaning in particular both domain and codomain objects are in the subcategory for any morphism in the subcategory, that $\text{Ob} \cat{C} = \text{Ob} \cat{D}$ because the identity arrows are isomorphisms, but isn't this putting the cart in front of the horse?
I imagine this is fairly basic, if there's no mention to the objects of the category, what does one generally assume they are? Thanks in advance for the clarification, let me know if I can clarify further.