Vinothkumar Raman

Thanks a lot :D

Now this clears things up

I think you are right. I was thinking about $Rel$ as a category of where morphisms are relations

That example is super confusing

And full criterion makes it have all the arrows too, but all of them are not symmetric

But my confusion is $id$ arrows are automatically symmetric and hence should be there in the subcategory it is defined to be reflector too and it will have all objects too, since they all have to have $id$ arrows which makes the subcategory have all the objects

Its definied as $\rho \cup \rho^{-1}$ in the reflector example and $\rho \cap \rho^{-1}$ in co-reflector. Not sure what it really is

There is an example about $Rel$ and $Sym$ categories

Yea, I kind of understood that after I posted it.

And yea its in Abstract and Concrete categories

So I am definitely missing something here

But the proposition goes over another iff condition which expects $A$ to be full sub category, which i dont find necessary.

My question was $id_X$ is definitely considered as B-morphism hence should also have a reflector in $A$, since $X \in A$ too. And the reflector could infact be just $id_X$ and $X$ itself.

But if I take your definition It doesnt say anything about subcategory at all.

Actually the definition is like this $A$ is a subcategory of $B$ and $f: X \rightarrow Y$ and $X$ is in B and $Y$ is in $A$ and $f$ factors through some $Z$ in $A$ for every $X$ and which is unique in $A$.

I mean $id_X$ is a reflection of $X$

I have a small question while reading Joy of cats but it seems too small to write a question. So the question is If $A$ is a subcategory of $B$ and $X \in A$ then $id$ should be reflection whether $A$ is full or not right? But I see a proposition which claims it is a reflection only when its full? Why is it so?

I mean $id_X$ is a reflection of $X$

I have a small question while reading Joy of cats but it seems too small to write a question. So the question is If $A$ is a subcategory of $B$ and $X \in A$ then $id$ should be reflection whether $A$ is full or not right? But I see a proposition which claims it is a reflection only when its full? Why is it so?