Thanks. I guess I can chase it, if there is no reference that makes it explicit.

Is the category of Coalgebras over a field a topos? I have found several conflicting views about this scouring the litrature. In the least i'm pretty sure that the yonneda embedding for the category of coalgebras has a left exact left adjoint. But then I saw on the nlab that there's a name for this condition: "lex total" and there's also a theorem stated there that the only reason for a lex total category to not be a topos is that the set of isomorphism classes of objects is too big...

But this would be a weird reason for Coalgebras to not be a topos

(my coalgebras are cocommutative)

@SaalHardali That adjoint preserves finite products, but I don't see that it preserves equalizers.

I take that back; I was thinking about cocommutative coalgebras. I have no intuition for general coalgebras.

Oops, you were also thinking about cocommutative coalgebras.

@WilliamBalderrama This might be the thing i'm overlooking! I admit i cannot see a reason for the left adjoint to preserve all finite limits. Do you have an example in mind though?

I don't, sorry. I'd be happy to learn one, if it exists.