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quotient category

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In the last list, point 1, it says:

$F(h\circ f)= F(h)\circ F(f) = F(h)\circ F(g) = F(h\circ g)$, as $F$ is a (contravariant) functor. However, $h\circ g$ is not defined unless $D = \operatorname{cod}(g)=\operatorname{dom}(h) = B$, which is impossible by the assumption that $B\ne D$.

I'm probably missing something obvious, but isn't $F(h)\circ F(g) = F(h\circ g)$ required for a functor only if $h \circ g$ is defined in the first place? Otherwise you could show that any functor is injective on objects: Assume $F(A_1) = B = F(A_2)$, then $id_B \circ id_B = F(id_{A_1}) \circ F(id_{A_2}) = F (id_{A_1} \circ id_{A_2})$, which is undefined unless $A_1 = A_2$.

Good point! I meant to come back and work on this entry some more... adding more examples, and discussing the difference between this definition and the one given in Mac Lane. I totally forgot about it! I will get to it later today. Thank you!

The entry is now in a better shape than it was a few days ago. Improvements will be on-going, as always. But unless there are more suggestions, I am going to leave it "as is" now.


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