TITLE: Yoneda's Lemma in Vakil's notes
QUESTION [8 upvotes]: Vakil's Notes
in the exercise 1.3Y, what does 1.3.10.2 commutes with the maps 1.3.10.1 mean? I can't see any relation between 1.3.10.1 and 1.3.10.2.

REPLY [8 votes]: 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 1.3.10.1 via $f$.  Note that in 1.3.10.2, 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 $B$).