[05WU]
Lemma 2.1.3 . Let F : ๐ โ ๐ F\colon\mathcal{C}\to\mathcal{D} F X : ๐ X / โ ๐ F ( X ) / F_{X}\colon\mathcal{C}_{X/}\to\mathcal{D}_{F\left(X\right)/}
[05WV]
Proof. Consider the commutative square
๐ X / \textstyle{\mathcal{C}_{X/}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} ๐ F ( X ) / \textstyle{\mathcal{D}_{F\left(X\right)/}\ignorespaces\ignorespaces\ignorespaces\ignorespaces} ๐ \textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces} ๐ . \textstyle{\mathcal{D}.}
The vertical functors and the bottom horizontal functor preserve pullbacks.
The right vertical functor is conservative. It follows that the top
horizontal functor preserves pullbacks as well.
โ