Lemma 4.31.7. Assumptions as in Lemma 4.31.6.

1. 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.

2. 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.

3. 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.

Proof. Let $(X, Y, \phi )$ and $(X', Y', \phi ')$ be objects of $\mathcal{X} \times _\mathcal {Z} \mathcal{Y}$. Set $Z = H(X)$ and identify it with $I(Y)$ via $\phi$. Also, identify $M(Z)$ with $F(L(X))$ via $\alpha _ X$ and identify $M(Z)$ with $G(K(Y))$ via $\beta _ Y$. Similarly for $Z' = H(X')$ and $M(Z')$. The map on morphisms is the map

$\xymatrix{ \mathop{\mathrm{Mor}}\nolimits _\mathcal {X}(X, X') \times _{\mathop{\mathrm{Mor}}\nolimits _\mathcal {Z}(Z, Z')} \mathop{\mathrm{Mor}}\nolimits _\mathcal {Y}(Y, Y') \ar[d] \\ \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(L(X), L(X')) \times _{\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M(Z), M(Z'))} \mathop{\mathrm{Mor}}\nolimits _\mathcal {B}(K(Y), K(Y')) }$

Hence parts (1) and (2) follow. Moreover, if $K$ and $L$ are equivalences and $M$ is fully faithful, then any object $(A, B, \phi )$ is in the essential image for the following reasons: Pick $X$, $Y$ such that $L(X) \cong A$ and $K(Y) \cong B$. Then the fully faithfulness of $M$ guarantees that we can find an isomorphism $H(X) \cong I(Y)$. Some details omitted. $\square$

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