Lemma 87.30.4. All of the separation axioms listed in Definition 87.30.1 are stable under composition of morphisms.
Proof. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of formal algebraic spaces to which the axiom in question applies. The diagonal $\Delta _{X/Z}$ is the composition
\[ X \longrightarrow X \times _ Y X \longrightarrow X \times _ Z X. \]
Our separation axiom is defined by requiring the diagonal to have some property $\mathcal{P}$. By Lemma 87.30.3 above we see that the second arrow also has this property. Hence the lemma follows since the composition of (representable) morphisms with property $\mathcal{P}$ also is a morphism with property $\mathcal{P}$. $\square$
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