Is tensor a left adjoint or a right adjoint?

Question

It is a well known fact that the tensor functor preserves limits. Therefore, by the adjoint functor theorem, it should be a right adjoint to some left adjoint functor (hom I'm guessing). However, wikipedia says that it is a left adjoint here. So I am wondering which one is correct.

Answer 1

For the sake of having an answer, and expanding my comments:

To be explicit, let's consider the tensor functor $(-) \otimes_R N$ from right $R$-modules to abelian groups, where $N$ is a left $R$-module. This functor is left adjoint to the hom functor $\text{Hom}_{\mathbb{Z}}(N, -)$, and therefore preserves colimits. It preserves limits if and only if $N$ is finitely generated projective; see this answer for a proof.

The well-known fact is that tensoring preserves direct limits. "Direct limits" are actually colimits, and in more modern terminology would be referred to as directed colimits, a special case of filtered colimits. The confusion here comes from the fact that "direct limit" is old terminology and actually predates the modern terminology of colimits and limits.

My preference here is to never use the terms "direct limit" or "inverse limit," precisely because they cause this confusion, and to instead use the modern terms "directed colimit" and "codirected limit."