# A variation of a codomain fibration

Let $$\mathcal C$$ be a category with pullbacks, let $$Arr(\mathcal C)$$ be its arrow category. The functor $$cod\colon Arr(\mathcal C)\to\mathcal C$$ that takes an object of $$Arr(\mathcal C)$$, i.e. a morphism in $$\mathcal C$$, to its codomain is a canonical example of a Grothendieck fibration. The cartesian arrows are exactly the pullback squares.

There is an easy generalization of this fact. Take any category $$\mathcal D$$ a functor $$F\colon D\to C$$ and consider the comma category $$id_{\mathcal C}\downarrow F$$. The objects of $$id_{\mathcal C}\downarrow F$$ are arrows $$g\colon c\to F(d)$$, the morphisms are pairs $$(f,h)$$, where $$f$$ is a morphism in $$\mathcal C$$, $$h$$ is a morphism in $$\mathcal D$$ and $$\require{AMScd} \begin{CD} c@>f>>c'\\ @VgVV@VVg'V\\ F(d)@>>F(h)>F(d') \end{CD}$$ commutes.

There is an obviously defined functor $$cod_F\colon id_{\mathcal C}\downarrow F\to\mathcal D$$ that takes $$g\colon c\to F(d)$$ to the $$d$$ object and a square like the one above to the $$h$$ morphism.

Note that $$id_{\mathcal C}\downarrow id_{\mathcal C}\simeq Arr(\mathcal C)$$ and that $$cod_{id_{\mathcal C}}\simeq cod$$.

Question: What is the appropriate reference for the fact that $$cod_F$$ is a fibration?

I need this fact in a paper I am writing, the paper is intended for an audience that will probably not accept that this is "easy to check" or something.

• Shouldn't this follow from the pullback stability of Grothendieck fibrations?
– asdq
Feb 17, 2020 at 12:42
• @ArnaudD. What do you mean? The functor $\mathrm{cod}_F$ is the strict pullback of $\mathrm{cod}$ along $F$.
– Pece
Feb 17, 2020 at 16:18
• Yes, this works fine. The comma category is naturally a lax pullback, but every lax pullback is constructed as a strict pullback of an arrow category. Feb 17, 2020 at 16:24
• @Pece Oh right, I hadn't thought of that. I was thinking about $\operatorname{cod}(F)$ as a "lax pullback" of $id_\mathcal{C}$ along $F$. Feb 17, 2020 at 17:52
• @asdq Yes, this is what I was missing. Thank you. If you convert your comment to an answer, I will be happy to accept it. Feb 17, 2020 at 20:42