Remark 56.3.10. Let $A$ and $B$ be rings. Let us endow $\text{Mod}_ A$ and $\text{Mod}_ B$ with the usual monoidal structure given by tensor products of modules. Let $F : \text{Mod}_ A \to \text{Mod}_ B$ be a functor of monoidal categories, see Categories, Definition 4.43.2. Here are some comments:

Since $F(A)$ is a unit (by our definitions) we have $F(A) = B$.

We obtain a multiplicative map $\varphi : A \to B$ by sending $a \in A$ to its action on $F(A) = B$.

Take $A = B$ and $F(M) = M \otimes _ A M$. In this case $\varphi (a) = a^2$.

If $F$ is additive, then $\varphi $ is a ring map.

Take $A = B = \mathbf{Z}$ and $F(M) = M/\text{torsion}$. Then $\varphi = \text{id}_\mathbf {Z}$ but $F$ is not the identity functor.

If $F$ is right exact and commutes with direct sums, then $F(M) = M \otimes _{A, \varphi } B$ by Lemma 56.3.1.

In other words, ring maps $A \to B$ are in bijection with isomorphism classes of functors of monoidal categories $\text{Mod}_ A \to \text{Mod}_ B$ which commute with all colimits.

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