Exactness and Products of Categories
A functor is left exact (resp. right exact) if it preserves finite limits
(resp. finite colimits). Let $\mathcal \otimes \colon \mathcal A \times
\mathcal B \longrightarrow \mathcal C$ be a bifunctor. Is there a
connection between left (or right) exactness of $A \otimes -$ and $-
\otimes B$ for all objects $A \in \mathcal A$ and $B \in \mathcal B$, and
left (or right) exactness of $\otimes$ as a bifunctor?
For example if $k$ is a commutative ring, is $- \otimes_k -$ exact if and
only if $M \otimes_k -$ and $-\otimes_k M$ exact for any $k$-module $M$?
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