In my category theory course, we saw the principle of adjunction and there is something that I really do not understand. Let me explain it with an example.

We were studying the hom-set relation, i.e. we define a map $$\theta_{N, Z}: {}_A\text{Hom}_B(M\otimes_B N, Z) \to {}_B\text{Hom}_C(N, {}_A\text{Hom}(M,Z))$$ where $A, B, C$ are $k$-algebra (for $k$ a field) and $M,N,Z$ are modules in ${}_A\text{Mod}_B, {}_B\text{Mod}_C, {}_A\text{Mod}_C$ respectively. Explicitly, we get $$\theta_{N,Z}(f)(n)(m) = f(m\otimes_B n)$$ for any $f \in {}_A\text{Hom}_B(M\otimes_B N, Z)$. This function is in fact a bijection and the inverse in given by $$\theta_{N, Z}^{-1}(g)(m\otimes_Bn) = g(n)(m).$$ From this fact, my teacher deduce that the functor $M \otimes_B -$ is the left adjoint of the functor ${}_A\text{Hom}(M,-)$. But here I don't really understand why we make a difference between left and right because the choice of "the sens" of $\theta_{N,Z}$ seems arbitrary, we could very well have chosen $\tilde{\theta}_{N, Z} = \theta_{N, Z}^{-1}$ and now $M \otimes_B -$ becomes the right adjoint. What's wrong with my argument? Why could we not take $\tilde{\theta}_{N, Z} = \theta^{-1}_{N, Z}$ ?


Be careful, the notion of "left" or "right" adjoint does not refer to the direction of the natural transformation, but rather where the functor is standing in the hom.

An adjunction between $F$ and $G$ is a natural isomorphism $\hom(F(X),Y)\cong \hom(X,G(Y))$. No matter the direction of the natural transformation, $F$ is the left adjoint because it is on the left in the hom (in $\hom(a,b)$, $a$ is on the left, $b$ is on the right) and $G$ the right adjoint because it is on the right in the hom.

So if you choose $\theta^{-1}$, $M\otimes_B-$ is still left adjoint.

Contrary to the direction of $\theta$, the position in the hom is not arbitrary and makes this notion slightly asymmetric. For instance left adjoints always preserve colimits, but not necessarily limits, whereas right adjoints always preserve limits but not necessarily colimits, so there is an actual distinction.

Note: it does happen that the tradition is often to write it as a natural isomorphism $\hom(F(X),Y)\overset\cong\to \hom(X,G(Y))$, so that's probably why you were confused, but again, the direction $\to$ does not really matter.


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