Vakil's Notes

in the exercise 1.3Y, what does commutes with the maps mean? I can't see any relation between and

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    $\begingroup$ Welcome to Maths SE. 2 things: For questions of this nature it is mandatory that you show some working or at least your thoughts on how to get to the solution, and secondly, if someone gives an answer that is useful to you don't forget to tick it. All the best. $\endgroup$ – BLAZE Dec 10 '15 at 7:25
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    $\begingroup$ Contrary to the downvotes and the comment above, this is a perfectly reasonable question with plenty of context provided. $\endgroup$ – Eric Wofsey Dec 10 '15 at 7:33

It means that given any two objects $B$ and $C$ and a map $f:B\to C$, the diagram $$\require{AMScd} \begin{CD} \operatorname{Mor}(C,A) @>{i_C}>> \operatorname{Mor}(C,A')\\ @V{}VV @V{}VV \\ \operatorname{Mor}(B,A) @>{i_B}>> \operatorname{Mor}(B,A') \end{CD}$$ commutes, where the vertical maps are given by via $f$. Note that in, you are supposed to have a map $i_C:\operatorname{Mor}(C,A)\to \operatorname{Mor}(C,A')$ for each object $C$; in this diagram we are using two of these maps (for $C$ and for $B$).


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