Lemma 4.31.7. Assumptions as in Lemma 4.31.6.

If $K$ and $L$ are faithful then the morphism $\mathcal{X} \times _\mathcal {Z} \mathcal{Y} \to \mathcal{A} \times _\mathcal {C} \mathcal{B}$ is faithful.

If $K$ and $L$ are fully faithful and $M$ is faithful then the morphism $\mathcal{X} \times _\mathcal {Z} \mathcal{Y} \to \mathcal{A} \times _\mathcal {C} \mathcal{B}$ is fully faithful.

If $K$ and $L$ are equivalences and $M$ is fully faithful then the morphism $\mathcal{X} \times _\mathcal {Z} \mathcal{Y} \to \mathcal{A} \times _\mathcal {C} \mathcal{B}$ is an equivalence.

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