Fix omission and pedantry

Thanks to Patrick Rabau and Luis
https://stacks.math.columbia.edu/tag/0013#comment-7762
https://stacks.math.columbia.edu/tag/0013#comment-7863

categories.tex

@@ -206,12 +206,11 @@ \section{Definitions}
 
 \begin{definition}
 \label{definition-subcategory}
-A {\it subcategory} of a category $\mathcal{B}$ is
-a category $\mathcal{A}$ whose objects and arrows
-form subsets of the objects and arrows
-of $\mathcal{B}$ and such that source, target
-and composition in $\mathcal{A}$ agree with those
-of $\mathcal{B}$. We say $\mathcal{A}$ is a
+A {\it subcategory} of a category $\mathcal{B}$ is a category $\mathcal{A}$
+whose objects and arrows form subsets of the objects and arrows of $\mathcal{B}$
+and such that source, target and composition in $\mathcal{A}$ agree with those
+of $\mathcal{B}$ and such that the identity morphism of an object of
+$\mathcal{A}$ matches the one in $\mathcal{B}$. We say $\mathcal{A}$ is a
 {\it full subcategory} of $\mathcal{B}$ if $\Mor_\mathcal{A}(x, y)
 = \Mor_\mathcal{B}(x, y)$ for all $x, y \in \Ob(\mathcal{A})$.
 We say $\mathcal{A}$ is a {\it strictly full} subcategory of $\mathcal{B}$
@@ -372,7 +371,7 @@ \section{Definitions}
 \end{lemma}
 
 \begin{proof}
-This lemma proves itself.
+Omitted.
 \end{proof}
 
 \begin{lemma}