The Stacks project mentions colimit in the Sets and introduces the following equivalence relationship: $m_{i} \sim m_{i^{'}}$ if $m_{i} \in M_{i}$, $m_{i^{'}} \in M_{i^{'}}$ and $M(\varphi)(m_{i}) = m_{i^{'}}$ for some $\varphi: i \rightarrow i^{'}$.
My question is, is this really an equivalence relationship? If we want to satisfy the symmetry of equivalence relationships, should there be $\exists \psi: i^{'} \rightarrow i$ s.t. $\psi \circ \varphi = Id$? Otherwise, how to meet the requirements that $m_{i} \sim m_{i^{'}} \Rightarrow m_{i^{'}} \sim m_{i}$?
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$\begingroup$ I notice that the chain of morphisms at the citation can be read left-to-right or right-to-left with the same semantics. $\endgroup$– Eric TowersCommented Jun 17 at 2:36
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3$\begingroup$ It says "the equivalence relation generated by $\sim$", which means the smallest equivalence relation which contains $\sim$. $\endgroup$– anankElpisCommented Jun 17 at 2:37
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$\begingroup$ @anankElpis Can the complete equivalence relationship be explicitly described？ $\endgroup$– jhzgCommented Jun 17 at 2:40
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1$\begingroup$ The link you give describes the relation in the sentence after the one you quoted. I don't know if you'd call that explicit but it's the best you'll get. $\endgroup$– anankElpisCommented Jun 17 at 2:47
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