Suppose that $\mathcal{C}$ is a complete symmetric monoidal closed category, and that $M$ is a monoid. According to nCatLab, the functor category $[M, \mathcal{C}]$ is also a symmetric monoidal closed category with the tensor given by the pointwise tensor product: $(F\otimes G)f := F(f) \otimes G(f)$.
I would like to know, in more concrete terms, what the internal hom in this category looks like (the adjoint of $\otimes$). To illustrate: $[M, \textsf{Set}]$ with $\otimes = \times$, can be more concretely described as the category of $M$-sets, and exponentials $B^A$ defined with the equivariant functions from $M\times A \to B$. What is the equivalent concrete representation when $\textsf{Set}$ is replaced with an arbitrary complete monoidal closed category?
The result stated on nCatLab is actually more general than the above, and works when $M$ is replaced by any small category. I would also like to know: do you actually need the assumption that $\mathcal{C}$ is complete in the present case where $M$ is a monoid?
(The argument is given in terms of Kan extensions which is a concept I'm not yet comfortable with yet, so I'd be grateful for an answer that doesn't presuppose that notion.)