Definition 4.32.1. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category over $\mathcal{C}$. A strongly cartesian morphism, or more precisely a strongly $\mathcal{C}$-cartesian morphism is a morphism $\varphi : y \to x$ of $\mathcal{S}$ such that for every $z \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$ the map

$\mathop{Mor}\nolimits _\mathcal {S}(z, y) \longrightarrow \mathop{Mor}\nolimits _\mathcal {S}(z, x) \times _{\mathop{Mor}\nolimits _\mathcal {C}(p(z), p(x))} \mathop{Mor}\nolimits _\mathcal {C}(p(z), p(y)),$

given by $\psi \longmapsto (\varphi \circ \psi , p(\psi ))$ is bijective.

Comment #3991 by Praphulla Koushik on

I am sure it is already known but wanted to say for some one who is reading for first time. What is called as strong cartesian morphism here is called as cartesian morphism in some places for example in Vistolis's descent theory notes (http://homepage.sns.it/vistoli/descent.pdf) page number 44, remark 3.2

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