diff --git a/derived.tex b/derived.tex
index 3efd21db..c1e5e6f9 100644
--- a/derived.tex
+++ b/derived.tex
@@ -1594,7 +1594,7 @@ \section{Localization of triangulated categories}
 
 \begin{lemma}
 \label{lemma-limit-triangles}
-Let $\mathcal{D}$ be a triangulated category.
+Let $\mathcal{D}$ be a pre-triangulated category.
 Let $S$ be a saturated multiplicative system in $\mathcal{D}$
 that is compatible with the triangulated structure.
 Let $(X, Y, Z, f, g, h)$ be a distinguished triangle in $\mathcal{D}$.
@@ -1723,11 +1723,11 @@ \section{Localization of triangulated categories}
 $(s_4, s'_4, s''_4) : (X_2, Y_2, Z_2, f_2, g_2, h_2) \to
 (X_3, Y_3, Z_3, f_3, g_3, h_3)$ in $\mathcal{I}$.
 We would be done if the compositions
-$X' \to X_1 \to X_3$ and $X' \to X_2 \to X_3$ where equal
+$X' \to X_1 \to X_3$ and $X' \to X_2 \to X_3$ were equal
 (see displayed equation in 
 Categories, Definition \ref{categories-definition-cofinal}).
 If not, then, because $X/S$ is filtered, we can choose
-a morphism $X_3 \to X_4$ in $S$ such that the compositions
+a morphism $X_3 \to X_4$ in $X/S$ such that the compositions
 $X' \to X_1 \to X_3 \to X_4$ and $X' \to X_2 \to X_3 \to X_4$ are equal.
 Then we finally complete $X_3 \to X_4$ to a morphism
 $(X_3, Y_3, Z_3) \to (X_4, Y_4, Z_4)$ in $\mathcal{I}$