This is probably a very basic question, because it seems to me that this is literally the point of an adjunction, but I don't really get it still. (note, I don't really get adjunctions in general).
if $F:D\to C$ is left adjoint to $G$ then there are two natural transformations
$\eta:FG\to 1_C$
$\gamma:1_D\to GF$
Why is it not $\gamma:GF\to 1_D$? It seems weird to me that the counit and unit would have opposite types, and I'm not sure why they would? I feel like not understanding this is the thing that's preventing me from understanding adjunctions.
It's because the unit $\eta$ comes from transposing the identity $Fx\to Fx$ across the adjunction to get a map $x\to GFx$ and the counit $\varepsilon$ comes from transposing the identity $Gy\to Gy$ to get a map $FGy\to y$. There's no way to use the adjunction to produce maps with the signatures you're wondering about: in an adjunction, you can turn an $F$ into a $G$ by moving it from left to right, and you can turn a $G$ into an $F$ by moving it from right to left, but the other two possibilities are not valid.