See [Fulton’s Intersection Theory, B.6.9.].
Lemma
If \(X\hookrightarrow Y \hookrightarrow Z\) are closed embeddings, then there is a canonical embedding \(\Bl_X Y \hookrightarrow \Bl_X Z\), and the exceptional divisor of \(\Bl_XZ\) restricts to the exceptional divisor of \(\Bl_XY\).