Let $(\mathscr{C}, \otimes , I)$ monoidal category, a monoid $(R, e_R, \mu )$ is a object $R \in \mathscr{C}$ with morphisms $e_R: I \to R$, $\mu: R \otimes R \to R$ with the well knowed unital e associative commutative diagram.

If $(\mathscr{C}, \otimes , I)$ has a symmetry (is enought a braid) then the tensor prodoct $M \otimes N $ of two objects, each one with a monoid structure, has a natural monoid structure, then we have a monoidal symmetrical (braided) category of monoid of $(\mathscr{C}, \otimes , I)$. Given a monoid $R$ a left $R$-module $(M, \mu)$ is a object $m \in C$ with a morphism $\mu: R \otimes M \to M $ with usual unitary and associative diagram, similarly are defined the right modules.

Now we suppose that $(\mathscr{C}, \otimes, [-, ?], I)$ ia a monoidal closed, symmetrical category, given a right $R$-module $M$and a a left $R$-module $N$, In analogy to the classical algebra context, let $[M, N]^R $ the kernel of the two morphisms gived by the compositions:

$[M, N] \xrightarrow{- \otimes R} [M \otimes R, N \otimes R ] \xrightarrow{[1, \cong ]} [M \otimes R , R \otimes N ] \xrightarrow{[1, \mu_N]}[R \otimes M, N ]$

$[M, N] \xrightarrow{ [\mu_M, 1] } [R \otimes M, N ]$

**How to prove that $[M, N]^R$ is a left $R$-module (for the structure inducted by $M$, and inducted by $N$) ?**

This is mentioned on §3 of the Stefan Schwede & Brooke Shipley article 'Algebras and modules in monoidal model categories'

http://arxiv.org/abs/math/9801082

I know that A. Kock studied the monoidal -closed structure of algebras of a (commutative) triple in monoidal closed symmetrical categories (then a much more general and complex result).

I ask where I can find a explicit but rigorous proof.